Rapper Missy Elliott’s hit 2002 song "Work It" (parental advisory warning required!)was my go-to LSAT study prep song. This was not due to my deep affinity for Elliott’s music (I’m more of a Childish Gambino type of girl), but because the lyrics of ‘Work It’ contain a hidden key to mastering contrapositive statements. In our post on necessary and sufficient conditions, we covered the basics of conditional statements. Now that we know our Ps and Qs, we are ready to manipulate conditionals, which will include learning about inverses, converses, and the dreaded contrapositives.

Introduction to Manipulating Conditional Statements

Remember that conditional statements are "if-then" statements, comprised of a hypothesis and a conclusion. In formal logic, these statements are written in the equation form of p → q (if p, then q). By interchanging and/or negating the hypothesis and conclusion, you can find the inverse, converse, and contrapositive of a conditional statement. As we go forward with manipulating conditionals, you’ll want to keep these "Work It" lyrics handy:

Is it worth it? Let me work it.

I put my thing down, flip it, and reverse it.

In these lyrics, Missy Elliott’s "thing' is represented by a conditional statement that, through manipulation, will be summarily flipped and reversed.

Inverses

In order to find the inverse of a conditional statement, you will need to use interchanging. Interchanging involves switching the hypothesis and conclusion so that the hypothesis becomes the conclusion and the conclusion becomes the hypothesis. So if the equation of a conditional is p → q (if p, then q), the equation of its inverse is q → p (if q, then p). Returning to Missy Elliott, putting your thing down means being given your initial conditional statement and to get the inverse you must flip (interchange) that statement.

Example One

Conditional statement:

• “If Erin eats croissants, she will be happy”
• Erin eats croissants → she’s happy
  

Inverse:

• “If Erin is happy, she will eat croissants”
• Erin is happy → she eats croissants

Example Two

Conditional statement:

• “If Vedika works hard, she will become a doctor”
• Vedika works hard → she’ll become a doctor
  

Inverse:

• “If Vedika becomes a doctor, she will work hard”
• Vedika becomes a doctor → she works hard

Warning: Interchanging to find the inverse does not mean that you can simply switch the order of which concept is presented first in a sentence. Conditional statements can be written with the conclusion presented before the hypothesis, such as the conditional “Pete is studying if he is at the coffee shop.” In this case, the conclusion is presented first (Pete is studying) and the hypothesis comes second (If he is at the coffee shop). To simply switch the order of those statements and say “If Pete is at the coffee shop then he is studying” does not actually interchange the hypothesis and the conclusion.

Converses

To find the converse of a conditional, we will keep the same order of the conditional but negate both the hypothesis and the conclusion. Negating means to take the opposite meaning of the hypothesis and conclusion. So if the equation of a conditional is p → q (if p, then q), the equation of its converse is not p → not q (if not p, then not q). Harkening back to the second part of our ‘Work It’ lyrics, once we put our thing down by getting our conditional statement, we can find the converse by reversing (negating) that statement.

Example One

Conditional statement:

• “If Erin eats croissants, she will be happy”
• Erin eats croissants → she’s happy
  

Converse:

• “If Erin does not eat croissants, she will not be happy”
• Erin doesn’t eat croissants → she’s not happy

Example Two

Conditional statement:

• “If Vedika works hard, she will become a doctor”
• Vedika works hard → she’ll become a doctor
  

Converse:

• “If Vedika does not work hard, she will not become a doctor”
• Vedika does not work hard → she doesn’t become a doctor
  

Pro Tip: Inverse and converse sound very similar. To keep them straight, just remember that the ‘i’ in inverse stands for interchange. If inverse means interchange, that means that converse must mean negation.

Contrapositives

Contrapositives require both interchanging and negating the hypothesis and conclusion of a conditional. So if the equation of a conditional is p → q (if p, then q), the equation of its contrapositive is not q → not p (if not q, then not p).Here’s where you can use all of Missy Elliott’s edifying lyrics at once - to get the contrapositive of a conditional, you need to put your thing (conditional) down, flip (interchange) it, AND reverse (negate) it.

Example One

Conditional statement:

• “If Erin eats croissants, she will be happy”
• Erin eats croissants → she’s happy
  

Contrapositive:

• “If Erin is not happy, she will not eat croissants”
• Erin is not happy → she does not eat croissants

Example Two

Conditional statement:

• “If Vedika works hard, she will become a doctor”
• Vedika works hard → she’ll become a doctor
  

Contrapositive:

• “If Vedika does not become doctor, she will not work hard”
• Vedika does not become a doctor → she does not work hard

Conclusion

As with learning the basics of conditional reasoning, the only real way to get the hang of manipulating conditionals is to practice them. Make sure you practice some examples where the initial conditional is not written in straightforward p → q format, where the hypothesis is written first. You can use our conditional “Pete is studying if he is at the coffee shop” to get you started on non-straightforward conditionals. So throw on some Missy Elliott (and put in your headphones, if folks with sensitive ears are around!) and start to "Work It."