Logic Laws: Converse, Inverse, Contrapositive & Counterexample

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Logical statements

Logical statements are utterances that can be tested for truth or falsity. The phrase, "Jennifer's white birds" is not a logical statement because it lacks meaning. The phrase, "Jennifer is best at magic" is not logical because it is an opinion; it is not testable. The phrase, "Jennifer wears dresses every Tuesday" is logical because it can be tested. Either she wears dresses on Tuesdays or she does not.

Humans are not born to be logical. Most humans do not begin to learn logic until they are around 10 years old. Logic is a learned mathematical skill, a method of ferreting out truth using specific steps and formal structures. Some of those structures of formal logic are converse, inverse, contrapositive, and counterexample statements.

Logical statements must be tested to be valid. For example, one of the two statements below is logical in that they can be tested for its truthfulness. One is an opinion, which cannot be tested for truthfulness:

• Cuban food tastes best.

• Jennifer is a man.

The first statement is an opinion and is neither logical nor factual; it cannot be tested to be true. We know the second statement can be tested for its truthfulness. The second statement is logical but not factual.

Logic statement examples

Which of these phrases or utterances is a logical statement? Remember, it need not be true, just testable.

1. Mint chocolate chip ice cream is delicious.

2. Jennifer is a woman.

3. 3 penguins and 2 water buffalo

4. Fricasé de Pollo is a type of Cuban food.

Statements 2 and 4 are logical statements; statement 1 is an opinion, and statement 3 is a fragment with no logical meaning.

Four testable types of logical statements are converse, inverse, contrapositive, and counterexample statements. They can produce logical equivalence for the original statement, but they do not necessarily produce logical equivalence.

Logical equivalence

Suppose instead of writing an opinion for our first statement and a logical but not factual second statement, we wrote:

• Jennifer is alive.

• Living women eat food.

We can use these statements to form a conditional statement with a hypothesis and a conclusion:

This type of if-then statement is the heart of logic. We can immediately see that the two statements result in a true conditional statement. Here is Jennifer; she is alive; she eats food (and we already know she likes Cuban food).

A logical equivalence recasts the same hypothesis and conclusion as a negative statement that produces the same result:

These statements have logical equivalence because they contain the same content and arrive at the same result. Statements with logical equivalence are either both true or both false.

Converse statements

The original if-then conditional statement was:

Switching the hypothesis for the conclusion provides the converse statement:

We have the same words, but the order of the two parts has changed. Has the truth of the conditional statement changed? In this case, the statement is still true, but it would not have to be true.

Switching the conclusion for the hypothesis does not automatically prove the logical conditional statement, so the converse statement could be true or false.

Inverse statements

A logical inverse statement negates both the hypothesis and the conclusion. Again, our original, conditional statement was:

By carefully making the hypothesis negative and then negating the conclusion, we create the inverse statement:

The inverse statement may or may not be true.

Let's compare the converse and inverse statements to see if we can make any judgments about them:

• Converse: If Jennifer eats food, then Jennifer is alive.

• Inverse: If Jennifer is not alive, then Jennifer does not eat food.

Both of those produce true statements. Neither would have to produce a true statement, but in this case they did. It is not possible for one to produce a true statement and the other to produce a false statement.

We now know these three facts about converse and inverse statements:

1. If one is true, the other statement is true.

2. If one statement is false, the other is false.

3. Converse and inverse statements are logically equivalent to one another.

Contrapositive statements

If the converse reverses a statement and the inverse negates it, could we do both? Could we flip and negate the statement?

Our original conditional statement was:

To create the logical contrapositive statement, we negate the hypothesis and the conclusion and then we also switch them:

If the conditional statement is true, then the logical contrapositive statement is true. If the logical contrapositive statement is false, then the conditional statement itself is also false. They have logical equivalence.

Counterexamples

If you can find a substitute that tests the logical validity of the statement (but not its factual accuracy), you know the claim is not always true and is therefore not logically valid.

We need only find one instance, called a counterexample, where the conditions set out in our arguments are not valid:

• Original statement: If Jennifer is alive, then Jennifer eats food.

• Converse: If Jennifer eats food, then Jennifer is alive.

• Inverse: If Jennifer is not alive, then Jennifer does not eat food.

• Contrapositive: If Jennifer does not eat food, then Jennifer is not alive.

We would need to find a single example of one of these conditions, any one of which would be a counterexample:

1. A living woman who does not eat food, or

2. A woman who eats food but who is not alive, or

3. A nonliving woman who eats food, or

4. A woman who does not eat food but who is alive

If we can find such an example, even a single example, in which the premises are valid but the conclusions are false, we would have a counterexample showing the original argument is invalid.

Surely you can see - leaving out zombies and vampires and other imaginary creatures - that we cannot produce a counterexample for any of our logical statements; our argument is valid.

The logical result of all this work with converse, inverse, contrapositive, and counterexample logical statements is, we learn that Jennifer is a living, breathing woman who eats.

And she likes Cuban food!