Conditional Reasoning: Contrapositive, Mistaken Reversal, Mistaken Negation
The contrapositive flips and negates both sides of a conditional statement, and it's always logically equivalent to the original. A mistaken reversal swaps the sufficient and necessary conditions without negating them, and a mistaken negation negates both sides without swapping them. Both are invalid, and they're actually contrapositives of each other.
One statement, 4 versions
| Version | Example | Valid? |
|---|---|---|
| Original | If I tutor in Manhattan, then I tutor in NYC | Given |
| Contrapositive | If I don't tutor in NYC, then I don't tutor in Manhattan | Valid |
| Mistaken reversal | If I tutor in NYC, then I tutor in Manhattan | Invalid |
| Mistaken negation | If I don't tutor in Manhattan, then I don't tutor in NYC | Invalid |
What is the contrapositive? What do mistaken reversal (converse) and mistaken negation (converse) look like on the LSAT?
Bite-sized Logical Reasoning arguments may seem impossible to understand, but they’re pretty manageable once you’ve got a grip on the basics.
In this article, I’ll share the basics of conditional reasoning with you.
Original statement
If I tutor the LSAT in Manhattan, then I tutor the LSAT in New York City.
Symbolized
Manhattan -> NYC
Mistaken reversal / converse (invalid)
If I tutor the LSAT in New York City, then I tutor the LSAT in Manhattan.
Symbolized
NYC -> Manhattan
False because this statement implies that I tutor in a different part of NYC (Brooklyn, Queens, Staten Island, or the Bronx).
Mistaken negation / inverse (invalid)
If I do not tutor the LSAT in Manhattan, then I do not tutor the LSAT in New York City.
Symbolized
Manhattan -> NYC
Again, this statement is false because I could be in another borough of NYC.
Contrapositive (valid)
If I do not tutor the LSAT in New York City, then I do not tutor the LSAT in Manhattan.
Symbolized
NYC -> Manhattan
This is true. It’s impossible for me to tutor the LSAT in Manhattan if I don’t tutor the LSAT in NYC because Manhattan is in NYC.
It’s worth noting the mistaken reversal and mistaken negation are the contrapositives OF EACH OTHER. They are logically equivalent. Why? Because they’re flawed for the same reason – they confuse necessary and sufficient conditions.
The sufficient condition
appears to the left of the arrow in the “symbolized” sections above
is often indicated by the words “if” and “when”
is enough to cause the necessary condition to follow, but it’s not necessarily required for the necessary condition to occur
serves as the evidence
The necessary condition
appears to the right of the arrow in the “symbolized” sections above
is often indicated by the words “then” and “must”
often appears after a comma
is required by the sufficient condition
serves as the conclusion
Why this is important
Breaking down which parts of the argument are sufficient and necessary allows you to determine the evidence and conclusion. This helps you figure out potential flaws and opportunities to strengthen/weaken the argument.
Further reading
Wikipedia’s article on the contrapositive is solid.
Frequently asked questions
What is a contrapositive on the LSAT?
The contrapositive negates and reverses a conditional statement's two parts. If the original is 'if A, then B,' the contrapositive is 'if not B, then not A.' It's always logically valid and equivalent to the original statement. It only makes sense once sufficient vs necessary conditions is solid, so start there if the arrow feels shaky.
What's a mistaken reversal on the LSAT?
A mistaken reversal swaps the sufficient and necessary conditions without negating them, turning 'if A, then B' into 'if B, then A.' That's an invalid inference, since the necessary condition doesn't guarantee the sufficient one.
What's a mistaken negation on the LSAT?
A mistaken negation negates both parts of a conditional without swapping their order, turning 'if A, then B' into 'if not A, then not B.' It's invalid for the same reason a mistaken reversal is: it confuses necessary and sufficient conditions. Negating a conditional is a different move from reversing it; I cover it in negating conditional statements.
How are mistaken reversal and mistaken negation related?
They're contrapositives of each other and logically equivalent, both flawed because they confuse necessary and sufficient conditions. You'll see both inside real assumption questions in arguments, contrapositives, and assumptions.
I've written free explanations for every question in LSAT PrepTest 159, 158, 157, 156, 155 and 154, with the right answer and why each wrong answer fails. All LSAT explanations are here.
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