Chapter 1: Reconstructing and analyzing arguments

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, Barack 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. If you can construct a  counterexample to an argument, the argument is invalid.

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 traveling 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.

                                                       

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.

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Exercise 5: Determine whether or not the following arguments are valid by using the informal test of validity. If the argument is invalid, provide a counterexample.