This month's Word of the Month is "contrapositive." If you've ever taken a course in geometry or logic, you've likely heard the word "contrapositive" before. One definition of contrapositive is: "The contrapositive of a conditional statement is formed by negating both the hypothesis and the conclusion, and then interchanging the resulting negations."
That's a reasonable definition, but if your logic course isn't fresh in your mind, it might not mean much. So let's break it down into pieces that are a little more manageable.
A conditional statement is a statement in the form: If ______ then _______. Here are a few examples:
If it is snowing, then the ground is white.
If x = 3 then x + 2 = 5.
If the clock strikes 13 then it's time to get your clock fixed.
In each of these conditional statements, the "if" part is called the hypothesis, and the "then" part is called the conclusion.
So the hypotheses are:
It is snowing.
x = 3.
The clock strikes 13.
The conclusions are:
The ground is white
x + 2 = 5
It's time to get your clock fixed.
With that in mind, we can determine the contrapositve of each statement. We swap the order of the hypothesis and the conclusion, and then write the opposite of each.
Let's look at the third example: If the clock strikes 13 then it's time to get your clock fixed.
The opposite of the hypothesis is "The clock does not strike 13" and the opposite of the conclusion is "It's not time to get your clock fixed." If we now swap the order of these in the original statement, we get the following:
If it's not time to get your clock fixed, then your clock did not strike 13.
It turns out that a conditional statement and its contrapositive are equivalent statements. In other words, if a statement is true, its contrapositive is also true, and if a conditional statement is false, its contrapositive is also false.
Can you find the contrapositives of the other two examples?
While you're trying to work out those contrapositives (the answers are posted below), it is interesting to break the word down into its component parts:
Contra = in contrast to
posit = to assume or argue
ive = pertaining to
So a contrapositive is a statement which is in contrast to an argued statement.
And finally, here are your other two contrapositives:
If the ground is not white then it's not snowing.
If x + 2 does not equal 5 then x does not equal 3.