Having spent a few months studying logic and proofs, I find that I pay closer attention to the statements people make in everyday conversation. One type of argument that I see misused is logical implication. Implications are conditional statements of the storied: “if statement-1, then statement-2” type (i.e. if p, then q). The statement following “if” is called the hypothesis and the statement following “then” is called the conclusion.
An implication can exist in four common forms: the given implication and its inverse, converse, and contrapositive. An implication and its contrapositive are logically equivalent, and can be trusted to yield sound reasoning. The converse and inverse of an implication are not reliable and may give false results. If an implication is true, then its contrapositive should be as well. However, the converse and inverse of an implication may or may not be true–this is how they can get one into trouble. Let’s look at a few examples.
Start with the sentence,
“All human beings are mammals.”
As an implication derived from this sentence could be stated as,
“If one is a human being, then one is a mammal.”
Below is this implication with its converse, inverse, and contrapositive. The format for each form is shown in Table 1 so that you can understand how they are formed. Examples of the initial implication and the three additional forms are provided in Table 2.
Table 1 – Forms
Implication: if Statement-1, then Statement-2
Contrapositive: if NOT Statement-2, then NOT Statement-1
Converse: if Statement-2, then Statement -1
Inverse: if NOT Statement-1, then NOT Statement-2
Table 2 – Examples
Implication: If one is a human being, then one is a mammal.
Contrapositive: If one is NOT a mammal, then one is NOT a human being.
Converse: If one is a mammal, then one is a human being.
Inverse: If one is NOT a human being, then one is NOT a mammal
Notice how both the implication and its contrapositive lead to true conclusions while the inverse and converse lead to false conclusions.
Here is an example in which all four forms are true.
Implication: If today is Friday, then tomorrow is Saturday.
Contrapositive: If tomorrow is NOT Saturday, then today is NOT Friday.
Converse: If tomorrow is Saturday, then today is Friday.
Inverse: If today is NOT Friday, then tomorrow is NOT Saturday.
As mentioned earlier, the converse and inverse may or may not be true, and for this reason, they cannot be trusted. The unreliability of the implication’s converse and inverse are easy to spot in this scenario because we are certain of truthfulness of the statements in the original implication. In everyday conversation, it may not be so straightforward because missing information can cloud the picture.
Consider this scenario: Flights are available from City A to City B on Mondays, Wednesdays and Fridays.
Implication: If today is Monday, then there is a flight to City B.
Contrapositive: If there is NOT a flight to City B, then today is NOT Monday.
Converse: If there is a flight to City B, then today is Monday.
Inverse: If today is NOT Monday, then there is NOT a flight to City B.
Someone who is only aware of the Monday flight would not likely accept the converse because the current day of the week is obvious; however, the inverse might seem completely reasonable.
Obviously, the greatest danger of making or being taken in by the converse or inverse occurs when all the facts are not available. Therefore, the safest thing to do is to assume that both the converse and inverse of any implication are false until proven otherwise. Of course, if the initial implication is false, then avoiding its converse and inverse will not do much good.
Unfortunately, in everyday life we are often encouraged to form questionable implications—especially by advertisers. Consider these statements and what they attempt to imply:
“I use X, and I have never felt better.”
“ We have thousands of happy customers.”
Mathematicians use logic to test arguments, but it can useful for the rest of us too. After all, 9/10 people who read this post are glad they did!