How do we show that an argument is invalid?

If we show that an argument is invalid, what do we do then?

Remember that an argument is valid if it is impossible for the premises to be true and the conclusion false at the same time.  To show that an argument is invalid, we must give an example of a possibility in which the premises could be true and the conclusion false at the same time.

Consider the following example:

Cassandra owns a Mercedes-Benz.  Rich people own Mercedes-Benz automobiles.   Therefore, Cassandra must be rich.

Is this argument valid?  Consider all the possibilities.  (I recognize that the term "rich" is vague.  This is the kind of sentence we would hear in everyday conversation.)  The second premise, "Rich people own Mercedes Benz automobiles," does not exclude the possibility that some people who are not rich own Mercedez-Benz automobiles (i.e., Cassandra could  have a rich uncle who bought the the car for her; Cassandra perhaps was once rich, but is now poor because she squandered all of her money on expensive automobiles and so who is now poor.)  If either of these possibilities is true, then both premises could be true, but the conclusion would be false.  Any argument in which it IS possible to have true premises and a false conclusion at the same time is invalid.

Please note  (and this is very important!!):  In showing how the argument could be invalid, we have not discussed the truth of any of the premises.  We did not deny the truth of premise one or two.  We have simply talked about what could or could not be the case in view of the premises stated.  We do not show an argument to be invalid by saying one of the premises is false.


Consider a variation on the same argument: 

Cassandra owns a Mercedes-Benz.   Only people who are rich own Mercedes-Benz automobiles.  Therefore Cassandra must be rich.

In this example, the second premise excludes the possibility of anyone but a rich person owning a Mercedes; the second premise rules out the possibility that owners of Mercedes-Benz cars inherit them from rich uncles or that they suddenly become poor.)   Hence, it is impossible for the premises to be true and the conclusion to be false.   Hence, the argument is valid, because if it is true that only rich people own a Mercedes-Benz, and if it is true that Cassandra owns one, then it is impossible for it to be false that Cassandra is rich.

Note:  Remember our discussion above about the first version of this argument?   Remember we said that Cassandra could  have a rich uncle who bought the the car for her; Cassandra perhaps was once rich, but is now poor because she squandered all of her money on expensive automobiles and so who is now poor.  These considerations do not demonstrate the invalidity of the argument.  They may show that the second premise is false.  Remember the actual truth or falsity of a premise does not affect whether or not it is valid.


Let's try another example:

All good teachers come to class on time.  Professor Simpson always comes to class on time.  Professor Simpson must be a good teacher.

Can the premises be true and the conclusion false at the same time.  Think about the first premise.   Even if we assume that it's true that all good teachers come to class on time, this claim does not exclude the possibility that some bad teachers also come to class on time.  There are many traits and behaviors that go into being a good teacher, i.e., knowing the subject matter, preparing for class, giving fair tests, as well as coming to class on time.  It is highly possible that there are some teachers who come to class on time, but who have none of the other traits of a good teacher.   Perhaps one such teacher is Professor Simpson: He comes to class on time, but his knowledge of the subject matter is minimal; he is never prepared for class; his tests are unfair.  The example of Professor Simpson shows us how it would be possible for the premises to be true and the conclusion false at the same time. 

Can you take the premises about good teachers and Professor Simpson and make a valid argument?  Write an answer in your notes and then click below to find the answer.   Be sure to try on your own before you check the correct answer!

Click to go to answer.

 


Consider the following example:

Everytime there is someone in the front yard, Spot bark.  Spot is barking, so someone must be in the front yard.

What about this one?  Is it valid?  Why or why not.   If it is invalid, rewrite it to make it valid.  Write an answer in your notes and then click below to check your answer.  Be sure to try on your own before you check the correct answer!

Click to go to answer.


Consider the following example.

Your grandmother had cancer, and your mother had cancer.  Therefore, you will get cancer.

You probably immediately recognize that this is not a valid argument.  Even if it is true that the person's mother and grandmother had cancer, this does not meant that it must be true that the person will also get cancer.  This is an invalid argument.

Yet, when you judge an argument to be invalid, you need to consider whether it might be strong.  Remember, there are varying degrees of strong.  An argument may be very strong or moderately strong. 

In this case, a family history of cancer is generally considered a factor that increases the risk of a person getting cancer.   If we revise our conclusion to say, "Hence, compared to other people without his family history, you have an increased likelihood of getting cancer," then this argument is moderately strong.   If the premises are true, then we have a good argument.

We cannot say the argument is very strong, because there are too many other variables that affect whether a person gets cancer or not.

In conclusion, to show that an argument is invalid, you must give an example of how the premises could be true and the premises false at the same time.

If an argument is invalid, ask if it could still be strong.