CHAPTER 9

VALID AND SOUND ARGUMENTS

9.1 VALIDITY AND SOUNDNESS

Validity is a most important concept in critical thinking. A valid argument is one where the conclusion follows logically from the premises. But what does it mean? Here is the official definition:

An argument is valid if and only if there is no logically possible situation in which the premises are true and the conclusion is false.

To put it differently, whenever we have a valid argument, if the premises are all true, then the conclusion must also be true. What this implies is that if you use only valid arguments in your reasoning, as long as you start with true premises, you will never end up with a false conclusion. Here is an example of a valid argument:

This simple argument is obviously valid since it is impossible for the conclusion to be false when the premise is true. However, notice that the validity of the argument can be determined without knowing whether the premise and the conclusion are actually true or not. Validity is about the logical connection between the premises and the conclusion. We might not know how old Marilyn actually is, but it is clear the conclusion follows logically from the premise. The simple argument above will remain valid even if Marilyn is just a baby, in which case the premise and the conclusion are both false. Consider this argument also:

Again the argument is valid—if the premises are true, the conclusion must be true. But in fact both premises are false. Some birds cannot fly (the ostrich), and bats are mammals and not birds. What is interesting about this argument is that the conclusion turns out to be true. So a valid argument can have false premises but a true conclusion. There are of course also valid arguments with false premises and false conclusions. What is not possible is to have a valid argument with true premises and a false conclusion. Here are some additional points about validity:

• A valid argument is one where it is logically impossible for the premises to be true and the conclusion to be false. But logically impossible does not mean “unlikely.” Consider this argument: Milton is a one-month-old human baby, and so Milton cannot walk. This seems cogent, but the argument is not valid because a one-month-old walking baby is not a logical impossibility. Imagine a scenario in which Milton is the product of a genetic experiment, and he is able to walk right after birth. Extremely implausible for sure, and maybe even biologically impossible. But the situation described is logically possible in the sense that there is no logical contradiction.1
• An argument that is not valid is invalid. This happens as long as there is at least one logically possible situation where its premises are true and the conclusion is false. Any such situation is known as an invalidating counterexample. It does not really matter whether the situation is realistic or whether it actually happens. What is important is that it is coherent and does not entail any contradiction. A single invalidating counterexample is sufficient to prove that an argument is invalid.
• Arguments are either valid or invalid, but we should not describe them as true or false. Because an argument is not a single statement, it is unclear what a true argument is supposed to be. Does it mean the argument has a true conclusion, or does it mean the argument is valid, or are we saying that the premises are true? It is confusing to speak of true and false arguments.

9.2 PATTERNS OF VALID ARGUMENTS

Valid arguments are useful because they guarantee true conclusions as long as the premises are true. But how do we know if an argument is valid? One indirect way is to see if we can come up with an invalidating counterexample. If we can, the argument is not valid. But of course, the weakness of this method is that when we fail to find a counterexample, this does not guarantee that the argument is valid. It is possible that we have not looked hard enough.

A more direct way of establishing validity is to demonstrate step by step how the conclusion of an argument can be derived using only logical principles. This is what formal logic is all about. But for everyday reasoning, a good understanding of some basic patterns of valid argument should also suffice.2

9.2.1 Modus ponens

Consider these two arguments:

Obviously, both arguments are valid. It does not matter whether the premises and the conclusions are true or not. Furthermore, these arguments are similar to each other in the sense that they have the same logical structure, which can be represented by this pattern:

Here, the letters P and Q are sentence letters. They are used to translate or represent statements. By replacing P and Q with appropriate sentences, we can generate the original valid arguments. This shows that the arguments have a common form. It is also in virtue of this form that the arguments are valid, for we can see that any argument of the same form is a valid argument. Because this particular pattern of argument is quite common, it has been given a name. It is known as modus ponens. But do not confuse modus ponens with the following form of argument, known as affirming the consequent:

Not all arguments of this form are valid. Here are two invalid ones:

Both arguments are invalid, and it is easy to find some invalidating counterexamples. For example, Putu might live in Jakarta and not Bali, and so the premises of the first argument are true but the conclusion is false. Similarly, it might be true that Chaak will buy you some roses if he loves you. But perhaps he will buy them even if he does not love you. Maybe he hates you so much that he decides to send you some roses that have been sprayed with a lethal virus.

9.2.2 Modus tollens

Modus tollens is also a very common pattern of valid argument:

Note that “not-Q” simply means the negation of Q—for example, “it is not the case that Q.” So if Q means “Superman has human DNA,” then not-Q would mean “it is not the case that Superman has human DNA,” or “Superman does not have human DNA.” But do distinguish modus tollens from the fallacious pattern of argument known as denying the antecedent:

9.2.3 Disjunctive syllogism

Both patterns are valid:

9.2.4 Hypothetical syllogism

Another pattern of valid argument:

9.2.5 Constructive dilemma

A pattern of valid argument with three premises:

When R is the same as S, this is an equally valid pattern:

9.2.6 Destructive dilemma

See if you can come up with your own example of this pattern of valid argument:

9.2.7 Reductio ad absurdum

Reductio ad absurdum is Latin for “reduced to absurdity.” It is a method for showing that a certain statement S is false:

If you can spot connections quickly you might notice that this is none other than an application of modus tollens. As an example, suppose someone claims that a human being’s right to life is absolute and so we should never kill or destroy human life. But is this acceptable? If it is, then it follows that when you are being attacked, it will be wrong for you to kill your attacker if this is the only way to prevent yourself from being harmed. But surely this is not correct. Most people would agree that in some situations when your life is threatened you can respond by deadly force, and this is recognized by our legal system. Since the original assumption leads to an absurd conclusion, this entails that the right to life is not absolute.

In mathematics, reductio proofs are also known as proofs by contradiction, or indirect proofs. Many well-known proofs, such as the proof that the square root of two is an irrational number, and Euclid’s proof that there are infinitely many prime numbers, employ this reductio method. They are beautiful proofs that are easy to understand. If you are interested you can look them up on the Internet quite easily.

9.2.8 Combining patterns to form more complex arguments

The patterns of valid arguments we have looked at are all rather simple. But they can be combined to form more complex arguments. This valid argument involves three applications of modus tollens:

When an argument gets even more complex, it might be a good idea to break it down into parts, or use a diagram known as an argument map to display the argument structure more clearly. (See chapter 11 for further discussion.)

9.3 ARGUMENTS INVOLVING GENERALIZATIONS

A generalization, or a general statement, is a statement that talks about the properties of a certain class of objects. In this chapter we shall be concerned only with the following three main kinds of generalizations:

Note that an existential generalization of the form “some F is G” means “at least one F is G.” In other words, the statement can be true even if there is just a single F that is G. The statement does not say that there are many Fs that are Gs. The reason to focus on such statements is that they are logically related to universal generalizations. The denial or negation of “every F is G” is “some F is not G.” For example, to show that “all politicians are corrupt” is false, all you need to find is one single politician who is not corrupt.

Another point to note about “some F is G” is that it does not logically imply “some F is not G.” Normally, if someone says “some vases are broken,” we might take him to imply that some vases are not broken. But this is not part of the literal meaning of the statement. After all, he can consistently maintain that all he knows is that some vases are broken, but he has no idea whether all of them are since he has not seen the rest of the vases.

One important aspect about the usage of universal generalizations is that “every F is G” is often used not as literally referring to every F in the world, but to some restricted class of Fs. For example, at a meeting, you might say something like “Everyone is here so let us begin.” But of course you are not really saying that everyone in the world is at the meeting. Rather, what you mean by everyone is something like “Everyone who is supposed to attend the meeting.” Similarly, when someone says that her apartment has been burgled and that “Everything is gone”, she is probably referring to every movable object of value inside the apartment, and it would be silly and unsympathetic to reply that the bathtub is still there.

Similarly, very often people use every to mean something like “most.” For example, it is often said that everyone loves children. But surely there are people who dislike children, perhaps thinking that they are noisy and naughty. Nevertheless, the claim is a harmless one as long as we do not take it literally. The problem comes when to avoid exceptions, general claims are qualified in such a way that they become vacuous. For example, when it is pointed out that not everyone loves children, someone might say that every normal person loves children. But what does normal mean? If normal just means loving children then the claim is indeed true but empty and normal becomes a weasel word.

9.3.1 Patterns of valid argument

Here are a few valid argument patterns involving every, with some examples:

But notice that all three arguments below are not valid:

The last argument on the right is one that lots of people get wrong. We can use a Venn diagram to see why it is not valid and construct a counterexample, taking the area of a region to be proportional to the number of items in the corresponding set. So for example, the diagram on the following page shows that there are more Gs and Hs than Fs. Furthermore, most Fs are Gs, and most Gs are Hs, but there is actually no F that is H! (An example: Most birds are creatures that can fly. Most creatures that can fly are insects. So most birds are insects.)

9.4 SOUNDNESS

Given a valid argument, all we know is that if the premises are true, so is the conclusion. But validity does not tell us whether the premises or the conclusion are actually true. If an argument is valid, and all the premises are true, then it is called a sound argument. Of course, it follows from such a definition that the conclusion of a sound argument must be true. An argument that is not sound is unsound.

In a discussion, we should try out best to provide sound arguments to support an opinion. The conclusion of the argument will be true, and anyone who disagree would have to show that at least one premise is false, or the argument is invalid, or both. This is not to say that we can define a good argument as a sound argument. (We shall discuss this issue in chapter 12.)

EXERCISES

1 This is not to say that the argument about Milton is a bad argument. Even though it is not valid, it is still inductively strong. Inductively strong arguments play a very important role in probabilistic and scientific reasoning, and we shall discuss them in more details in chapter 10.

2 But if you want to learn some formal logic, you can start with the online tutorials on sentential and predicate logic on our companion website. These are the two most basic systems of formal logic.