VALIDITY AND SOUNDNESS
VALIDITY The main goal of critical reasoning is to evaluate arguments: that is, to determine whether a given argument is a good one or a bad one.. The first steps toward this goal are to locate the conclusion, find the premises, and recognize the structure of the argument. Our efforts so far have been directed to those tasks. It’s time now to discuss the question of evaluation. Two main factors make an argument good or bad: (1) the relationship between the premises and the conclusion and (2) the status of the premises. Given this, we will introduce two concepts for evaluating arguments. The first, validity, has to do solely with the relationship between the premises and the conclusion. The second, soundness, concerns both the relationship between the premises and the conclusion and also the status of the premises. Let’s first concentrate on validity, which concerns only the relationship between the premises and the conclusion To understand the concept of validity, we need to realize that the purpose of an argument is to present a reliable form of reasoning, and reliability involves truth. Ideally, we want arguments that have the following feature: if we start with true premises, they should lead us to a true conclusion. In other words, we want our arguments to be truthpreserving. An argument that is truth-preserving in this way is called a valid, or a deductively valid, argument. Logicians often say that a valid argument is one in which the conclusion follows logically from (or is logically implied by) its premises. But what does this mean? Here is a more informative definition of validity: A valid argument is one whose premises are related to its conclusion in such a way that if all its premises were to be true, then its conclusion would have to be true. This definition calls attention to several important features of a valid argument. (1) As we said above, validity concerns only the relationship of the premises to the conclusion. In particular, validity is not determined by whether the premises and/or the conclusion are true or false. (2) The definition does not require that all of the premises actually be true. Rather, it invites you to consider what would happen if the premises were to be true. (3) If the premises of a valid argument were true, then the conclusion would have to be true as well. A valid argument might have false premises. But, if its premises were true,
then its conclusion could not possibly be false. The premises, if true, guarantee the conclusion. However, the definition does have a drawback: it is so “wordy” that it’s hard to remember. Another definition, which means exactly the same thing, is better. It will be our preferred definition.
Definition. An argument is valid if and only if it is logically impossible for its conclusion to be false when all of its premises are true.
It follows from our definition of validity that any argument that is not valid is invalid and vice versa: both of these terms are “all or nothing.” If an argument cannot guarantee the truth of its conclusion on the basis of the truth of its premises, it is simply invalid. There is no such thing as an argument that is somewhat valid, or mostly invalid. VALIDITY AND LOGICAL FORM How do we tell when an argument is valid? An argument is valid when it exemplifies a valid form of argument. By a valid form of argument we mean a pattern such that any argument with that form or following that pattern exactly will automatically be valid. To examine the form of arguments more easily, with fewer distractions, we will frequently substitute symbols or letters for actual words, phrases, or statements. This allows us to look at the form completely apart from the specific claims made by the premises. We can show, for example, that any argument of the form All A’s are B’s All B’s are C’s Therefore, All A’s are C’s is valid. That will tell us that whenever we substitute things for A, B, and C that make the premises true, the conclusion will be true too. For example, All collies are dogs. All dogs are animals. Therefore, all collies are animals. Of course, if our substitutions make one or both of the premises false, then anything can happen – the conclusion might be true, or it might be false, even though the argument is still valid. All of the following examples are valid arguments, even if they have a false premise or a false conclusion. They are all valid because they all follow exactly the same valid pattern.
A. All whales are fish. All fish are cold-blooded. Therefore, all whales are cold-blooded. B. All whales are fish. All fish live in water. So, all whales live in water.
(False) (True) (False) (False) (True) (True)
C. All whales are fish. (False) All fish suckle their young. (False) Therefore, all whales suckle their young. (True) D. All whales are mammals. All mammals suckle their young. So, all whales suckle their young. (True) (True) (True)
Each of these arguments is valid, even though three of them have at least one false premise, and one has a false conclusion. Each argument is valid because if all its premises were true, its conclusion would also have to be true; all four arguments exemplify the same valid form of argument. Argument D illustrates the fact that if a valid argument does have all true premises, the conclusion must also be true. On the other hand, if one or more of the premises of a valid argument is false, the conclusion might be true or it might be false; there is no guarantee either way. How do we show that an argument is invalid? This can be difficult, since many invalid arguments have true conclusions. The crucial point is to prove that even if all the premises were true, the conclusion could still be false. For that reason, paying attention to the form of the argument can help us. So we begin by learning how to identify an invalid form of argument. Since an invalid form of argument leaves open the possibility that true premises can lead to a false conclusion, we can show that a form is invalid by constructing an example of an argument following that form in which all the premises are true and the conclusion is false. We call this giving a counter-example to the argument. Consider example E: E. If the president resigned yesterday, then the vice-president has become president. The president did not resign yesterday, and so the vice-president has not become president. Although E may appear to be valid, it is actually invalid, and this can be discovered by examining its form. The form of argument E is: If p, then q Not p So, not q
We can show that this form is invalid by substituting statements for p and q that make all the premises true and the conclusion false, as in example F: F. If Dianne Feinstein is the President, then she is a U.S. citizen. Feinstein is not the President. So, she is not a U.S. citizen. Given that Dianne Feinstein is a U.S. citizen, but not the President, F conclusively shows that in arguments of this form, the truth of the premises does not guarantee the truth of the conclusion. Therefore, this is an invalid form of argument. Any argument that exemplifies a valid form of argument is a valid argument. Things are not quite as simple with invalid forms of argument, since one argument may exemplify many different forms. An argument is invalid only if all the forms it exemplifies are invalid. In practice, however, the number of forms that might be valid for any given argument is extremely limited, and we can readily see which forms are worth testing. If we find that these forms allow us to construct arguments with true premises and a false conclusion, then we have effectively shown that the argument is invalid. In most cases we cannot tell if an argument is valid just by noting whether its premises and its conclusion are actually true. A valid argument can have any combination except all true premises and a false conclusion. An invalid argument can have any combination whatsoever: all true premises and a true conclusion, all true premises and a false conclusion, some false premises and a false conclusion, or some false premises and a true conclusion. Thus, a false conclusion can be the result of either a valid argument or an invalid one. The key to discovering if it is valid is to ask whether it is ever possible for all the premises to be true and the conclusion false. If it is possible, then the argument is invalid. Only when that possibility is ruled out is the argument valid. SOUNDNESS The second concept, soundness, builds on the idea of validity. A sound argument must satisfy two criteria: it must be a valid argument, and all its premises must be true. Definition. An argument is sound if and only if it is valid and all of its premises are true. If either of these criteria is not met, the argument is unsound. This definition tells us two things. Like validity, soundness does not admit of degrees – an argument is either sound or it is unsound. Second, a sound argument always has a true conclusion. To determine whether an argument is sound (in contrast to determining its
validity) we must evaluate both its form and the truth of its premises. For most of this course we will concentrate on validity.