The purpose of this article is to introduce LD debaters to a few basic elements of formal logic; later articles will explore some applications of logic to the practice of debate.

Some people say that high school debate should be conducted and judged on the basis of logic alone-sola ratione, so to speak. Anyone acquainted with logic as a deductive science will have a hard time figuring out what these people mean. If they mean simply that debaters should avoid being illogical, well and good-but then who has ever suggested that debaters should be illogical? If they mean that the resources of logic should be sufficient to settle the truth of any debate resolution, they are betraying a basic confusion about the power of logic. Trying to generate substantive truths using only logic would be like trying to generate money using only arithmetic. Hopefully the readers of this article will come to understand why.

I. Preliminary Notions
Despite its limitations, logic is an indispensable tool in debate. Logic is the science of arguments. An argument is a set of sentences, one member of which is the conclusion and the others of which are premises offered in support of the conclusion. An inductive argument is one in which the premises are intended to make the conclusion likely or probable. For example,

• (1) Most Alabamians own guns.(premise)
• (2) Pat is an Alabamian.(premise)
• (3) Therefore, Pat owns a gun.(conclusion)

If both premises of this argument are true, they provide evidence supporting the conclusion, but they do not render the conclusion certain. Perhaps Pat is a member of the tiny Alabama chapter of Handgun Control and believes it is immoral to own guns.

A deductive argument, by contrast, is one in which the premises are intended to make the conclusion certain. For example,

• (4) No wombats are mammals.(premise)
• (5) J.S. Mill is a wombat.(premise)
• (6) So J.S. Mill is not a mammal.(conclusion)

If both premises of this argument are true, they make the conclusion not merely probable, but absolutely certain.

Obviously anyone who wants to argue for a given conclusion will prefer a deductive to an inductive argument when the former is available. Most constructive arguments in LD can be represented as deductive, and deductive arguments are the only sort of arguments which we can assess with a formal precision akin to that of mathematics. Therefore, we will focus exclusively on deductive arguments.

Astute readers will have noticed that the sample deductive argument above-about J.S. Mill-contains a glaring problem: Both of its premises and its conclusion are false. And yet it is as good logically as an argument can be. What gives? The argument is valid. That means that if the premises are true, the conclusion must be true. To determine that the argument is valid, we do not have to know whether the premises really are true or not. We can simply recognize that if the premises were true, the conclusion would also have to be true. So if no wombats are mammals and if J.S. Mill is a wombat, then J.S. Mill is not a mammal. But the argument is not sound. A sound argument is one which is valid and in which all the premises are true. This means that all sound arguments are valid (by definition), but not all valid arguments are sound (because some valid arguments have one or more false premises). Given the definition of validity, you can see that a sound argument must also have a true conclusion.

Please note that only an argument can be valid or invalid, sound or unsound; sentences or propositions do not possess these properties. Likewise, only sentences or propositions can be true or false; arguments cannot be true or false.

Obviously someone making an argument wants a sound argument, not merely a valid argument. But logic can deliver only validity. That is, an argument may be as logical as you please but still have a false conclusion. Logic tells us what propositions (conclusions) follow from the propositions (premises) we already believe, but it cannot tell us which propositions to accept as premises in the first place. Garbage in, garbage out, as they say. This is not just your author's opinion, or the majority view among professional logicians; nor is it a prediction or a generalization. It is the sober and indisputable truth about logic.

Already, you should begin to see the problem with the view that any interesting debate could be settled on the basis of logic alone. The only way a debate might be settled by logic alone is if both sides agreed to the same premises and the dispute were simply over the validity of the inference from those premises to competing conclusions. But debates are almost never like that-the debaters argue to different conclusions from different sets of premises. Often both sides present valid arguments, and the real question is, whose premises should be believed? For better or worse, logic by itself cannot answer this question.

II. Three Valid Forms
We said above that an argument has the property of validity when its conclusion must be true if its premises are true. But how can you ensure that your own arguments have this valuable property? Arguments are valid or invalid on the basis of their form, not on the basis of their content. In other words, it doesn't matter what the sentences in the argument are about or whether they are true or false. As long as the sentences are declarative and are properly arranged (more on this below), the argument they compose will be valid. The bad news is that there are literally an infinite number of valid argument forms. The good news is that we can represent most of the deductive arguments used in LD with only three simple forms.

To focus your attention on these forms, it will help to introduce a few symbols. Let us use the italicized letters p, q, r, and s to stand for declarative sentences. We will use the tilde (~) written before a sentence to signify the negation of that sentence. So if I assert ~p, then I am claiming that the declarative sentence p is false. Finally, let us symbolize conditional (if-then) sentences using an arrow (4) with the appropriate component sentences written at either end of the arrow. Conditional sentences are complex sentences-that is, they are composed of two simpler sentences. The first of these component sentences comes after the "if" and is called the antecedent. The second component sentence comes after the "then" and is called the consequent. Consider, for example, this conditional sentence: If it rains, then I will skip the game. The antecedent of this sentence is "it rains," and the consequent is "I will skip the game." If p stands for "it rains" and q stands for "I will skip the game," then we can represent the conditional sentence as: p--q.

Now we can introduce the three valid argument forms we need. The first is called modus ponens (MP). A modus ponens argument is of the form:

• p--q (premise)
• p (premise)
• So q (conclusion)

Here is an example of a valid MP argument:

• (7) If capital punishment deters crime, then capital punishment is moral.(premise)
• (8) Capital punishment deters crime.(premise)
• (9) Therefore, capital punishment is moral.(conclusion)

Even if you believe that (7) or (8) is in fact false, you can see that the argument is valid: if (7) and (8) are true, then (9) must also be true.

It may or may not be obvious to you that the order of the premises makes no difference to the validity of the argument. We could have switched (7) and (8), and (9) would just as surely follow. However, the order in which you present the premises of an argument can be very important to your audience's grasp and acceptance of the argument, and we will return to this subject briefly in a future article.

The second crucial argument form is modus tollens (MT), which is:

• p--q (premise)
• ~q (premise)
• So ~p (conclusion)

In the abstract, modus tollens is less obviously valid to most people than is modus ponens. But in fact, every MT argument is valid, and we use this pattern of reasoning all the time. For example,

• (10) If all human life were sacred, then it would be wrong to defend the innocent with lethal force.(premise)
• (11) But it's not wrong to defend the innocent with lethal force.(premise)
• (12) So not all human life is sacred.(conclusion)

Again, the order of the premises in an MT argument does not matter. So long as one premise is of the form p4q and the other premise is of the form ~q, the conclusion ~p necessarily follows.

The third and final valid argument form we need is called hypothetical syllogism (HS):

• p--q (premise)
• q--r (premise)
• So p--r (conclusion)

Unlike modus ponens and modus tollens, every sentence in a hypothetical syllogism is a complex conditional sentence-there are no simple p's or q's as premises or conclusion. Here is an example:

• (13) If pornography demeans women, then pornography communicates ideas.(premise)
• (14) If pornography communicates ideas, then pornography is a form of speech.(premise)
• (15) So if pornography demeans women, then pornography is a form of speech.(conclusion)

You should memorize the three argument forms we have learned so far. Your ability to apply these forms to more complex debate arguments will depend on your ability to recognize the simpler forms instantly.

Before learning how to apply MP, MT, and HS to more complex arguments, we should note two common fallacies or invalid forms which resemble the valid forms above. What makes a form of reasoning fallacious rather than valid? Simply that its premises even if true do not guarantee the truth of its conclusion. One common fallacy is denying the antecedent:

• p--q (premise)
• ~p (premise)
• So ~q (conclusion)

The best way to show that a given form of reasoning is fallacious is to produce a counter-example to it-an example of the form in question which has obviously true premises but an obviously false conclusion. For instance:

• (16) If pine trees are pink, then pine trees are colored. (premise)
• (17) Pine trees are not pink.(premise)
• (18) So pine trees are not colored.(conclusion)

Both premises are true, but the conclusion is false-pine trees are green, and therefore colored. The moral: Don't deny the antecedent!

The second fallacy you should take special care to avoid is called affirming the consequent, and it looks like this:

• p--q (premise)
• q (premise)
• So p (conclusion)

We can use the same first premise about pine trees to demonstrate the invalidity of this form:

• (19) If pine trees are pink, then pine trees are colored.(premise)
• (20) Pine trees are colored.(premise)
• (21) So pine trees are pink.(conclusion)

Here we have true premises but a false conclusion, proving that the argument is fallacious. Don't affirm the consequent!

With these warnings out of the way, we are ready to combine the simple argument forms into more complex patterns. First, instances of HS can be stacked one upon another, and validity will be preserved. If we have as premises

• p--q
• q--r
• r--s,

then we can validly conclude

• So p--s.

Think about it. We can apply HS to the first two premises to conclude p--r, and we can apply HS a second time to this conclusion and the third premise to get p--s. In fact, logically speaking, it doesn't matter how many conditionals we stack up. We can apply HS as many times as necessary to collapse a stack of properly related conditionals into a single conditional conclusion. (To be properly related, the conditionals must be such that the consequent of one conditional is the antecedent of another, whose consequent is the antecedent of another, and so forth.)

Further, we can combine instances of MP or MT with instances of HS. So, e.g., from the premises

• ~r
• p--q
• q--r

we can validly conclude

• ~p.

How does this work? Again, we are simply combining premises two-at-a-time using MP, MT, or HS to reach subconclusions, which we then combine with remaining premises using further applications of MP, MT, or HS. In this case, we applied MT to the first and third premises, which yielded ~q as a subconclusion. Then we combined this subconclusion with the second premise in another application of MT to reach the final conclusion ~p.

Instead of applying MT twice, we could just as easily (and validly!) have applied HS to the second and third premises to yield p--r and then applied MT once to this subconclusion and the first premise to yield the same final conclusion ~p.

From what we have said so far, you may already be able to see that we can combine the two forms of complex arguments to produce valid arguments of any length. If we let p* stand for a sentence different from p, we can use the premises

• p
• p--q
• q--r
• r--s
• s--p*
• p*4q*
• q*4r*
• r*4s*

to conclude

• s*.

We do this by applying MP repeatedly, beginning with the first two premises and continuing down the chain, or (if you prefer) by applying HS repeatedly to the second through eighth premises and then applying MP once to the first premise (p) and final subconclusion (p4s*).

III. Practice
And that's it. If you have followed me this far and have learned the three argument patterns explained above, you know enough formal logic to be an excellent debater. Of course, knowing this (or any) amount of formal logic does not guarantee that you'll be an excellent debater. You need to know how to apply your theoretical understanding of argument forms to real arguments. Later articles will discuss the application of the logical theory outlined here to LD cases and rebuttals. For now, you need to begin to recognize valid (and invalid) logical forms when you encounter them in spoken and written arguments. Arguments, as you know, do not typically present themselves as numbered lists of premises and conclusions. Instead, they come cloaked in prose, and you must learn to probe for the logical skeleton beneath the flesh of prose.

Here is an example of a typical written argument, the kind you might encounter (albeit at greater length) in an LD case:

• American colleges ought not to practice race-based affirmative action because doing so will lead to massive human suffering. Affirmative action will make colleges more appealing to students from other countries because it will increase their chances for admission. When more foreign students graduate from American colleges, more foreigners will stay in the U.S. and compete for high-paying jobs. This will put Americans out of work, which will destroy our economy. The collapse of our economy would destabilize the world politically, triggering a nuclear war. And we must do everything we can to prevent a nuclear war.
  
  

You'll notice that there is not a single if-then sentence in this paragraph. But we can easily translate the paragraph into a form which more clearly displays its validity:

• (22) Affirmative action will increase the chances of foreign students for admission to U.S. colleges. (premise)
• (23) If affirmative action will increase the chances of foreign student for admission to U.S. colleges, then affirmative action will make U.S. colleges more appealing to foreign students.(premise)
• (24) If affirmative action will make U.S. colleges more appealing to foreign students, then affirmative action will cause more foreign students to enroll in and graduate from U.S. colleges.(premise)
• (25) If affirmative action will cause more foreign students to enroll in and graduate from U.S. colleges, then affirmative action will cause foreign students will stay in the U.S. after graduation and compete with U.S. citizens for jobs.(premise)
• (26) If affirmative action will cause more foreign students will stay in the U.S. after graduation and compete with U.S. citizens for jobs, then affirmative action will cause U.S. citizens to lose their jobs.(premise)
• (27) If affirmative action will cause U.S. citizens to lose their jobs, then affirmative action will destroy the U.S. economy.(premise)
• (28) If affirmative action will destroy the U.S. economy, then affirmative action will destabilize the world politically.(premise)
• (29) If affirmative action will destabilize the world politically, then affirmative action will cause nuclear war.(premise)
• (30) If affirmative action will cause nuclear war, then affirmative action will cause massive human suffering.(premise)
• (31) If affirmative action will cause massive human suffering, then American colleges ought not to practice affirmative action.
• (32) So American colleges ought not to practice affirmative action.(conclusion)

If you symbolize this argument using p's and q's, you'll see that is it almost identical in form to the long argument which concluded with s* above.

If you compare my premise-conclusion restatement of the argument with the original paragraph version, you'll see that the conclusion and the last premise of the argument are both expressed by the first sentence of the paragraph version. The next-to-last premise, that nuclear war causes massive human suffering, is never stated outright in the paragraph, but we added it to our formalization in order to ensure the argument's validity. Sometimes there are good rhetorical reasons to leave premises unstated, but for the purposes of mastering the logic of arguments, you should work to make every assumption explicit. Much of the best philosophy consists in identifying and challenging premises which are usually taken for granted.

So how good is the affirmative action argument? Well, if we accept the formalization as suggested, the argument is logically impeccable. It is valid-the conclusion follows necessarily from the premises-and that is all we can demand logically from any argument. But logical or not, the argument stinks. Although the argument's moral (or normative) premise (31) is plausible, the argument also relies on a string of eight empirical (or descriptive) premises (22-29), all of which must be true for the argument to reach its announced conclusion. These empirical premises are not at all plausible. Perhaps there is good evidence supporting a few of them, but most look wildly exaggerated, and taken together, they are surely indefensible. Because of HS, premises (22)-(29) together entail that if affirmative action increases college admission chances for foreign students, then it will cause nuclear war. No one who understood all the words in that conditional could believe it, and no smart debater would want to have to defend such a far-fetched premise in order to attack affirmative action. But for now, the point is that despite its obvious weaknesses, the argument is logically valid, and any substantive debate about the argument will be decided not by logic alone, but rather by non-logical empirical or normative disputes about the truth of the argument's premises.

You can find arguments to analyze anywhere people are making arguments. Try looking at the editorial page or watching the talking heads on the news channels. Rarely will you find an argument presented as explicit instances of MP, MT, or HS. Your challenge is to translate the natural language of the arguer into the somewhat stilted if-then premises you have learned to check for validity. You will almost always have to add one or more premises which are only implicit in the argument's original presentation.

If you hear what you know is an argument but have no idea where to begin in mapping it, first find the conclusion. If the conclusion is best represented as a conditional, you know all the premises will be conditionals. If the conclusion is not a conditional, then you should expect at least one simple premise-represented by a non-conditional such as p or ~q-to make up part of an MP or MT. Work backwards from the conclusion through the chain of premises until you arrive at the starting point.

Finally, keep in mind that the three argument forms presented above, while adequate to represent the vast majority of, and maybe all, LD case arguments, are only a small fraction of the standard argument forms available in modern formal logic. Many real-life arguments, and some LD arguments, will have to be shoe-horned quite awkwardly to fit into these patterns, and some arguments will not fit at all. Fortunately, there are logical tools available to represent almost any deductive argument. Unfortunately, the Rostrum is not the place (and I am not the author) to acquaint you with that full set of tools. You may know other valid argument forms which allow you to represent this or that argument more gracefully than you could using only MP, MT, and HS. You should use all the logical tools at your disposal when you approach an argument. If you find the notion of a logical calculus scintillating, you should take a formal logic course in college, or even pick up a logic textbook and teach yourself.1 The sequels to this article will presuppose only a familiarity with MP, MT, and HS, but everything they say will apply to all valid arguments, regardless of form or complexity.(2)

1 One clear logic textbook is Virginia Klenk's understanding Symbolic Logic. New textbooks are prohibitively expensive; consider buying a used copy of an older edition online. 2 Thanks to Eric Barnes for comments on an earlier draft of this article.