Contrapositive
The contrapositive of an implication 'If A then B' is 'If not B then not A', which is logically equivalent to the original. Understanding contrapositive helps reframe arguments equivalently and avoids confusion with the converse or negation.
THE IDEA
CONTRAPOSITIVE
I believe that a great deal of logical power comes from flexibility, and this comes from being able to see things from different but equivalent points of view. Understanding how negation interacts with implication gives us a way of doing that. For example, believing that
“If some black people are better off than a white person then that person does not have white privilege.”
is equivalent to believing this:
“If you have white privilege then you are better off than all black people.”
The second statement is logically equivalent to the first. This means that the second one follows from the first, but also the first one follows from the second, so they are logically interchangeable (but they might still give slightly different emphasis to us non-logical humans.) For example, suppose I tell you:
If you travel abroad you must have a passport.
This is logically equivalent to saying:
If you don’t have a passport you can’t travel abroad.
However, they are perhaps emotionally a bit different – the first version is more about what you need to do in order to travel whereas the second is more about what you can’t do without a passport. These are two slightly different points in human terms although they’re logically equivalent.
These pairs of equivalent implications are called “contrapositives”. Formally, the contrapositive is a new statement that you can make, starting from an implication
A → B
The contrapositive statement is then
B is false → A is false
and is logically equivalent to the original statement. This means that whenever the original is true, the contrapositive is true. And whenever the original is false, the contrapositive is false. This is not to be confused with the converse, which is
B → A
and is logically independent of the original. It is also not to be confused with what you get if you just negate A and B individually:
A is false → B is false
which is the contrapositive of the converse, and so equivalent to the converse.
Negating A and B individually is the mistake people are making when they look at the statement
If you are a US citizen then you can legally live in the US.
and turn it into
If you are not a US citizen then you can’t legally live in the US.
Finally, the contrapositive is not to be confused with the negation:
¬(A → B)
In classical propositional logic, this negation is true exactly when A is true and B is false. It identifies a counterexample to the implication.
When to use it
In logical reasoning, when you're stuck proving a conditional, you can try proving its contrapositive instead.
In mathematics, many proofs use contrapositive to simplify implications.
In everyday arguments, understanding contrapositive helps you avoid logical fallacies like denying the antecedent or affirming the consequent.
In programming, use a condition and its contrapositive to check whether two logical requirements express the same constraint.
FROM THEORY TO EVERYDAY LIFE
In practice
White Privilege Argument
The statement 'If some black people are better off than a white person then that person does not have white privilege' is equivalent to its contrapositive 'If you have white privilege then you are better off than all black people.'
Passport Requirement
The statement 'If you travel abroad you must have a passport' is equivalent to its contrapositive 'If you don’t have a passport you can’t travel abroad.'
A MOMENT TO REFLECT
Put it to work.
How can you use the contrapositive to test the logical consistency of an argument?
Think of a conditional statement you often hear. What is its contrapositive? Does it seem as plausible?
Why is it important to distinguish contrapositive from converse when evaluating arguments?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)