1.6: Validity
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So far we have discussed what arguments are and how to determine their structure, including how to reconstruct arguments in standard form. But we have not yet discussed what makes an argument good or bad. The central concept that you will learn in logic is the concept of validity. Validity relates to how well the premises support the conclusion, and it is the golden standard that every argument should aim for. A valid argument is an argument whose conclusion cannot possibly be false, assuming that the premises are true. Another way of putting this is as a conditional statement: A valid argument is an argument in which if the premises are true, the conclusion must be true. Here is an example of a valid argument:
1. Violet is a dog
2. Therefore, Violet is a mammal (from 1)
You might wonder whether it is true that Violet is a dog (maybe she’s a lizard or a buffalo—we have no way of knowing from the information given). But, for the purposes of validity, it doesn’t matter whether premise 1 is actually true or false. All that matters for validity is whether the conclusion follows from the premise. And we can see that the conclusion, Violet is a mammal, does seem to follow from the premise, Violet is a dog. That is, given the truth of the premise, the conclusion has to be true. This argument is clearly valid since if we assume that “Violet is a dog” is true, then, since all dogs are mammals, it follows that “Violet is a mammal” must also be true. As we’ve just seen, whether or not an argument is valid has nothing to do with whether the premises of the argument are actually true or not. We can illustrate this with another example, where the premises are clearly false:
1. Everyone born in France can speak French
2. Barack Obama was born in France
3. Therefore, Barak Obama can speak French (from 1-2)
This is a valid argument. Why? Because when we assume the truth of the premises (everyone born in France can speak French, Barack Obama was born in France) the conclusion (Barack Obama can speak French) must be true. Notice that this is so even though none of these statements is actually true. Not everyone born in France can speak French (think about people who were born there but then moved somewhere else where they didn’t speak French and never learned it) and Obama was not born in France, but it is also false that Obama can speak French. So we have a valid argument even though neither the premises nor the conclusion is actually true. That may sound strange, but if you understand the concept of validity, it is not strange at all. Remember: validity describes the relationship between the premises and conclusion, and it means that the premises imply the conclusion, whether or not that conclusion is true. In order to better understand the concept of validity, let’s look at an
example of an invalid argument:
1. George was President of the United States
2. Therefore, George was elected President of the United States (from 1)
This argument is invalid because it is possible for the premise to be true and yet the conclusion false. Here is a counterexample to the argument. Gerald Ford was President of the United States but he was never elected president, since Ford Replaced Richard Nixon when Nixon resigned in the wake of the Watergate scandal. 2 So it doesn’t follow that just because someone is President of the United States that they were elected President of the United States. In other words, it is possible for the premise of the argument to be true and yet the conclusion false. And this means that the argument is invalid. If an argument is invalid it will always be possible to construct a counterexample to show that it is invalid (as I have done with the Gerald Ford scenario). A counterexample is simply a description of a scenario in which the premises of the argument are all true while the conclusion of the argument is false.
In order to determine whether an argument is valid or invalid we can use what I’ll call the informal test of validity . To apply the informal test of validity ask yourself whether you can imagine a world in which all the premises are true and yet the conclusion is false. If you can imagine such a world, then the argument is invalid. If you cannot imagine such a world, then the argument is valid. Notice: it is possible to imagine a world where the premises are true even if the premises aren’t, as a matter of actual fact, true. This is why it doesn’t matter for validity whether the premises (or conclusion) of the argument are actually true. It will help to better understand the concept of validity by applying the informal test of validity to some sample arguments.
1. Joan jumped out of an airplane without a parachute
2. Therefore, Joan fell to her death (from 1)
To apply the informal test of validity we have to ask whether it is possible to imagine a scenario in which the premise is true and yet the conclusion is false (if so, the argument is invalid). So, can we imagine a world in which someone jumped out of an airplane without a parachute and yet did not fall to her death? (Think about it carefully before reading on.) As we will see, applying the informal test of validity takes some creativity, but it seems clearly possible that Joan could jump out of an airplane without a parachute and not die—she could be perfectly fine, in fact. All we have to imagine is that the airplane was not operating and in fact was on the ground when Joan jumped out of it. If that were the case, it would be a) true that Joan jumped out of an airplane without a parachute and yet b) false that Joan fell to her death. Thus, since it is possible to imagine a scenario in which the premise is true and yet the conclusion is false, the argument is invalid. Let’s slightly change the argument, this time making it clear that the plane is flying:
1. Joan jumped out of an airplane travelling 300 mph at a height of 10,000 ft without a parachute
2. Joan fell to her death (from 1)
Is this argument valid? You might think so since you might think that anyone who did such a thing would surely die. But is it possible to not die in the scenario described by the premise? If you think about it, you’ll realize that there are lots of ways someone could survive. For example, maybe someone else who was wearing a parachute jumped out of the plane after them, caught them and attached the parachute-less person to them, and then pulled the ripcord and they both landed on the ground safe and sound. Or maybe Joan was performing a stunt and landed in a giant net that had been set up for that purpose. Or maybe she was just one of those people who, although they did fall to the ground, happened to survive (it has happened before). All of these scenarios are consistent with the information in the first premise being true and also consistent with the conclusion being false. Thus, again, any of these counterexamples show that this argument is invalid. Notice that it is also possible that the scenario described in the premises ends with Joan falling to her death. But that doesn’t matter because all we want to know is whether it is possible that she doesn’t. And if it is possible, what we have shown is that the conclusion does not logically follow from the premise alone. That is, the conclusion doesn’t have to be true, even if we grant that the premise is. And that means that the argument is not valid (i.e., it is invalid).
Let’s switch examples and consider a different argument.
1. A person can be President of the United States only if they were born
in the United States.
2. Obama is President of the United States.
3. Kenya is not in the United States.
4. Therefore, Obama was not born in Kenya (from 1-3)
In order to apply the informal test of validity, we have to ask whether we can imagine a scenario in which the premises are both true and yet the conclusion is false. So, we have to imagine a scenario in which premises 1, 2, and 3 are true and yet the conclusion (“Obama was not born in Kenya”) is false. Can you imagine such a scenario? You cannot. The reason is that if you are imagining that it is a) true that a person can be President of the United States only if they were born in the United States, b) true that Obama is president and c) true that
Kenya is not in the U.S., then it must be true that Obama was not born in Kenya. Thus we know that on the assumption of the truth of the premises, the conclusion must be true. And that means the argument is valid. In this example, however, premises 1, 2, and 3 are not only assumed to be true but are actually true. However, as we have already seen, the validity of an argument does not depend on its premises actually being true. Here is another example of a valid argument to illustrate that point.
1. A person can be President of the United States only if they were born in Kenya
2. Obama is President of the United States
3. Therefore, Obama was born in Kenya (from 1-2)
Clearly, the first premise of this argument is false. But if we were to imagine a scenario in which it is true and in which premise 2 is also true, then the conclusion (“Obama was born in Kenya”) must be true. And this means that the argument is valid. We cannot imagine a scenario in which the premises of the argument are true and yet the conclusion is false. The important point to recognize here—a point I’ve been trying to reiterate throughout this section—is that the validity of the argument does not depend on whether or not the premises (or conclusion) are actually true. Rather, validity depends only on the logical relationship between the premises and the conclusion. The actual truth of the premises is, of course, important to the quality of the argument, since if the premises of the argument are false, then the argument doesn’t provide any reason for accepting the conclusion. In the next section we will address this topic.
Exercise
Determine whether or not the following arguments are valid by using the informal test of validity. If the argument is invalid, provide a counterexample.
1. Katie is a human being. Therefore, Katie is smarter than a chimpanzee.
2. Bob is a fireman. Therefore, Bob has put out fires.
3. Gerald is a mathematics professor. Therefore, Gerald knows how to teach mathematics.
4. Monica is a French teacher. Therefore, Monica knows how to teach French.
5. Bob is taller than Susan. Susan is taller than Frankie. Therefore, Bob is taller than Frankie.
6. Craig loves Linda. Linda loves Monique. Therefore, Craig loves Monique.
7. Orel Hershizer is a Christian. Therefore, Orel Hershizer communicates with God.
8. All Muslims pray to Allah. Muhammad is a Muslim. Therefore, Muhammad prays to Allah.
9. Some protozoa are predators. No protozoa are animals. Therefore, some predators are not animals.
10. Charlie only barks when he hears a burglar outside. Charlie is barking. Therefore, there must be a burglar outside.
2 As it happens, Ford wasn’t elected Vice President either since he was confirmed by the Senate, under the twenty fifth amendment, after Spiro Agnew resigned. So Ford wasn’t ever elected by the Electoral College—as either Vice President or President.