Conditional Logic

Conditional Logic Intro

Conditional logic involves two terms that have a very specific relationship with one another. We call them the sufficient and the necessary conditions, respectively. Consider the following statement:

"If it's rainy, then it's cloudy."

Take a minute to think about it. Which of the above terms is sufficient for (guarantees) the other, and which is needed for the other?

The answer is that rain is sufficient for clouds, and clouds are necessary for rain. After all, rain guarantees that it's cloudy out, right? And you need clouds, or else you can't have rain.

Translating Conditional Logic

In the section above I presented you with a conditional relationship in "if, then" form. This is one of the most common ways of presenting conditional logic and can be translated as "if sufficient, then necessary." Just like in the previous section when we said "if it's rainy (sufficient), then it's cloudy (necessary)."

Here are some common ways that conditional logic is presented on the LSAT and how to translate them. Read through and start to familiarize yourself with the indicator words:

If sufficient then necessary
All indicates sufficient: All dogs are mammals

Dog -> Mammal

This works for "Any" too: Any dog is a mammal
Only indicates necessary: Only practice leads to mastery of the LSAT

LSAT Master -> Practice

This works for Only if too: Only if you practice will you master the LSAT

The majority of the time for this indicator word you will see Only or Only if. However, an exceptional case to be aware of is The Only. Consider the following:

"The only way to master the LSAT is practice."

Being an LSAT Master is still sufficient to know that a person practiced, and practice is still necessary for mastery. But now "master" (the sufficient condition) is closer to "only" - what gives? Technically, what follows "only" is "way" - which in this case refers to practice as being the "way" to mastery. So "only" still technically indicates the necessary condition, but the sufficient condition is closer to our indicator word. So make sure to consider the meaning of the sentence as a whole if you encounter "the only".

Unless indicates necessary: I won't go to the store unless I need milk

Store -> Need milk

I'm not going to the store unless I need milk - ok, so the only scenario in which I'm going to the store is if I need milk. I need to need milk to go to the store, so needing milk is the necessary condition. And if you see me at the store, what do you know for sure? That I must have needed milk! So me being at the store is sufficient to know that I needed milk.

This works for Until too: We won't take off until the pilot gets here

Take off -> Pilot here

Note: you negate the part before unless/until to get the sufficient condition

"No" can be complicated, but usually indicates sufficient: No dogs are reptiles Dog ->!Reptile

 Note: you end up negating both terms in the example above

 You could also see "Not" as an indicator word: Dogs are not reptiles               Dog ->!Reptile

You may also see examples like:

 Being in New Jersey guarantees that you are in the United States:       NJ -> USA

 Anyone from the boss' country club can get the job:                           CC -> Able to get the job

 I feel rested whenever I get a good nights sleep:                               Good nights sleep -> Well rested

 Humans never have 2 heads:                                                           Human -> !2 Heads

 Construction workers need to wear helmets on the job site:                CW on job site -> Helmet

Lastly, "if and only if" indicates both sufficient and necessary:

 I have a vegetarian diet if and only if I don't eat meat:     Vegetarian -> No meat and No meat -> Vegetarian

Diagramming

If you've studied the LSAT at all you may have seen someone diagram conditional logic whether in a course, curriculum, or explanation of a question. Diagramming can be a useful way to organize conditional logic in written form. The basic template is as follows:

"Sufficient -> Necessary"

So for the statement "If it's rainy, then it's cloudy," the diagram would look like:

Rain -> Clouds

Because rain is sufficient and clouds are necessary.

A note about diagramming: I typically discourage my students from diagramming. It's not necessary for top percentile LSAT scores (I got a 179 without even really knowing how to do it), and it adds complexity and takes up time on an already complicated, time-constrained test. However, I do use it as a teaching aid because it can be an illustrative way to organize and display conditional explanations.

The Contrapositive

Imagine a day where it's not cloudy. Could it possibly be raining?

Of course not! After all, clouds are necessary for rain. So there's a logical conclusion to be made if you don't have the necessary condition of a conditional relationship - you must also not have the sufficient.

So if we consider the conditional relationship:

Rain → Clouds

Then another logical statment follows:

~~Clouds~~ → ~~Rain~~

The fancy, technical term for this idea is "the contrapositive." What we've done in the diagram above is negated the necessary condition, negated the sufficient condition, and reversed their order.

A much easier way to think about it though is just that we don't have the necessary condition. And clouds are the necessary condition because they're needed for rain. And if we don't have what we need for rain, then it follows that it won't rain.

Chaining Conditionals

We know that if it's rainy, then it's cloudy. What if I introduce another conditional relationship, can we make any inferences by combining the two? Consider this:

"If it's cloudy, I won't get a tan."

Try to translate this into conditional logic - what's sufficient, and what's necessary?

Clouds are sufficient for no tan. So the diagram would be:

Clouds → ~~tan~~

Now we have a statement with clouds as a necessary condition, and a statement with clouds as the sufficient condition. We can "chain", or link those statements together as follows:

Rain → Clouds

Clouds → ~~Tan~~

Rain → ~~Tan~~

This makes sense, right? If it's rainy it's cloudy, and therefore I won't get tan. The contrapositive also follows, if I do get tan then it must not be cloudy and also must not be rainy:

Tan → ~~Rain~~