Conditional Reasoning: Mistaken Negations and Reversals

[Jon Denning]

A student of ours who's working through the PowerScore Logical Reasoning Bible asked a common question the other day, and I want to share it, and my response, with you.

Specifically she's been struggling with Mistaken Negations and Mistaken Reversals in conditional reasoning, and asked if I could help her better understand those two errors.

Here's my reply:

Conditional reasoning is a tricky concept for a lot of people, so don’t be too concerned if you struggle when you first begin studying it. That’s pretty normal. The good news is that once you crack it it becomes an extremely helpful skill, as you see it in about 15% of LR questions and about 40% or so of Games.

Here’s where I think the majority of students get hung up. What we really care about on the LSAT is what’s true: what we can know, what we can prove, with total certainty. In conditional reasoning, there’s very little that can actually be known with that level of confidence. Essentially there’s the original relationship that you’re told is true, and there’s the contrapositive (where you flip the conditions around the arrow and negate—make opposite—what they originally were). And that’s it!

So Mistaken Negations and Mistaken Reversals, both of which I’ll explain in greater detail below, are “Mistaken” because they present relationships that aren’t certain. They might be true, but that’s not good enough on the LSAT.


So let’s take a conditional statement, one that you’ve perhaps seen before in one of our books or courses, and see what we can do with it:

“If you get an A+, then you must have studied.”

First, we need to recognize that this is in fact conditional. How do we know? Because we’re given two events/phenomena—the A+ and the Study—that are related in an absolute, constant way, such that one (the sufficient condition) will tell you with certainty something about the other, and one (the necessary condition) is required for the other to exist.

Which is which? This is important since it will determine where we place them relative to their connecting arrow. And there are two questions you can ask to help make it clear. One is “which of these conditions tells me the other must have happened (or not happened, as is sometimes the case)?” The answer to that question will be your sufficient condition, and appear in front of the arrow in your diagram.

Here the A+ tells you that you studied, so we get A+ in front of our arrow and Study after it: A+ --> Study. That diagram means that when we see someone receive an A+, the sufficient condition in front of the arrow, we can follow the arrow and arrive at a truth: they studied, the necessary condition.

The other question you could ask yourself is “which of these conditions is required by the other in order for it to happen or exist?” Here, for an A+ to even be possible, studying is required (it MUST happen), so Study becomes the necessary condition and appears at the end of your arrow: A+ --> Study. Of course, that’s the same diagram as we just made...a good thing since that’s the correct relationship and we want to be consistent!

(Note that for this example both questions were fairly easy to answer and the relationship was pretty clear, but having two ways to determine sufficient vs necessary is still incredibly useful!; some scenarios will be much more easily identified by one of the conditions, sufficient or necessary, than the other, so having a two-fold means of analysis ensures you'll never be fooled)

At this point things are looking great! We’ve spotted conditional reasoning, identified the conditions, and labeled/diagrammed them correctly. So far so good. But what else can we do with that A+ --> Study relationship?

As I mentioned previously, beyond the original statement there’s only one absolute truth we can infer, and it’s known as "the contrapositive." To create it, take the original necessary condition, Study in this case, and negate it: Not Study. This now goes in front of the arrow: Not Study -->. Now take the original sufficient condition, A+ here, and negate that, too: Not A+. This goes after the arrow on your new diagram: Not Study --> Not A+. That means that if someone doesn’t study, we can be certain they don’t get an A+. Makes sense, as study was said to be required (necessary) for an A+, and if we ever lose something that’s required (if we know Not Study), then the thing that required it cannot happen (so Not A+). That’s why contrapositives are true!

But what about Mistaken Negations and Mistaken Reversals? Both commit essentially the same error, but in slightly different ways. It works like this: if all you know is that the original sufficient condition doesn’t happen (so if we know someone didn’t get an A+), or if all you know is that the original necessary condition did happen (so if we know someone studied), then you CANNOT make any further inferences and you have to stop! There’s nothing that either of those scenarios—sufficient missing or necessary present—will tell you with certainty, thus trying to draw conclusions from them is a serious error.

For the Mistaken Negation, it begins with the original sufficient condition gone. In this case, someone doesn’t get an A+ (Not A+). But is that enough to prove anything with certainty about whether that person studied? NO! An A+ means you studied, but a B or C- or F tells us nothing: someone could study for years and still fail, after all. Not A+ --> Not Study is the Mistaken Negation of our original terms, where the negated sufficient (Not A+) is used, incorrectly, to try to determine something about the necessary. So the rule is that if you don’t see the original sufficient condition—that someone got an A+--stop right there; any inferences you try to make without the sufficient are wrong.

For the Mistaken Reversal, a similar idea is in play but with a different trigger: we see the original necessary condition. Here, it would be if we knew that someone studied. What would that tell us about her grade? Not a thing! Again, someone could study their whole life and still not get that A+, so thinking that Study --> A+ is true is a critical mistake (and the most commonly-tested one on the LSAT with this stuff, turns out). The rule here, an LSAT mantra really, is that “Necessary conditions tell you nothing!” You cannot, ever, EVER go backwards against the arrow. It’s only when the necessary is missing that we know something, as that’s the start of the contrapositive.

Sufficient missing? STOP. Necessary present? STOP.

Let’s try one more (quick) example to hopefully drive it home. Suppose I tell you that to drive a car you need fuel. Notice, absolute language (“need”), two conditions (drive and fuel), and a relationship we can diagram: Drive --> Fuel. That means that according to my statement any time you see someone driving their car you can KNOW that the car has fuel in it. Simple enough. (Another sidenote: I've used an example that is, for the most part, real-world true...but don't assume the test makers will! Conditional reasoning doesn't have to follow commonsense rules, so be careful about diagramming based solely on what your experience tells you is the case)

The contrapositive is also pretty straightforward: If No Fuel, then No Drive. Diagrammed it would be something like No Fuel --> No Drive. This makes sense. If someone is out of gas, then they’re not driving that car (again, at least according to the statement I told you was true; obviously in the real world electric cars exist, but the real world doesn’t matter on the LSAT...just stick to what you're told is true and work from there).

But what about if you look around and see a parked car, that is a car that isn’t being driven? That’s No Drive, a missing sufficient. Does that mean for sure that it’s out of gas? Of course not. I’m not driving at the moment, but my car is safely parked nearby with a full tank. That’s the Mistaken Negation idea at work, and you can see why it’s an error—it tells you a relationship that may not be true.

And what if you have fuel in your car, does that mean for sure you’re driving it? That’s simply the necessary condition, Fuel, happening, but is it enough to know that car is being driven? Again, of course not. My car is full of gasoline, and sitting perfectly undriven in my garage right now. This is the Mistaken Reversal, and hopefully you can see why it too is a mistake. Fuel tells you nothing for sure.

So that’s a somewhat brief but hopefully illuminating look at conditional reasoning, as well as its two biggest traps—Mistaken Reversals and Mistaken Negations—in a nutshell.

I hope it helps! If so, and certainly if not, please let me know here, or reach out to us at (800) 545-1750 for more information.

Jon