When writing mathematical proofs, a number of things are important. One must be careful and precise with language. We must know what is known, either through axioms or other theorems, and what it is that we are trying to prove. Above all, we must be careful with our chain of logic.
Each step in the proof should flow logically from those that precede it. This means that if we do not use correct logic, we will end up with flaws in our proof. It is important to recognize valid logical arguments as well as invalid ones. If we recognize the invalid arguments then we can take steps to make sure that they don’t show up in our proofs.
One logical error or logical fallacy is called the converse error. This is a very common error, but one that can be hard to spot if we read a logical argument at a superficial level. Examine the following
If I eat fast food for dinner, then I have a stomachache in the evening. I had a stomachache this evening. Therefore I ate fast food for dinner.
Although this argument may sound convincing, it is logically flawed and constitutes an example of a converse error.
Definition of a Converse Error
To see why the above example is a converse error we will need to analyze the form of the argument. There are three parts to the argument:
- If I eat fast food for dinner, then I have a stomachache in the evening.
- I had a stomachache this evening.
- Therefore I ate fast food for dinner.
- If P, then Q.
- Therefore P.
Suppose we know that “If P then Q” is a true conditional statement. We also know that Q is true. This is not enough to say that P is true. The reason for this is that there is nothing logically about “If P then Q” and “Q” that means P must follow.
It may be easier to see why an error occurs in this type of argument by filling in specific statements forP and Q. Suppose I say “If Joe robbed a bank then he has a million dollars. Joe has a million dollars.” Did Joe rob a bank?
Well, he could have robbed a bank. But “could have” does not constitute a logical argument here. We will assume that both of the sentences in quotations are true. However, just because Joe has a million dollars does not mean that it was acquired through illicit means. Joe could have won the lottery, worked hard all his life or found his million dollars in a suitcase left on his doorstep. Joe’s robbing a bank does not necessarily follow from his possession of a million dollars.
Explanation of the Name
There is a good reason why converse errors are named such. The fallacious argument form is starting with the conditional statement “If P then Q” and then asserting the statement “If Q then P.” Particular forms of conditional statements that are derived from other ones have names, and the statement “If Q then P” is known as the converse.
A conditional statement is always logically equivalent to its contrapositive. There is no logical equivalence between the conditional and the converse. It is erroneous to equate these statements. Be on guard against this incorrect form of logical reasoning. It shows up in all sorts of different places.