Advanced Conditional Reasoning

There is nothing truly advanced about this chapter. I just could not think of a better name for it. We are building on the skills from the previous chapter, so if you passed that drill, you are perfectly ready for what follows.

Chaining Conditionals and Inferences

Let's build on our running example:

If I go to Harvard Law, then I would have taken the LSAT. And I would have to do a reading comprehension section if I had taken the LSAT.

There are two conditional statements here. Diagramming them:

Harvard Law Took the LSAT (1)

Took the LSAT Done a RC section (2)

Notice how "Took the LSAT" appears on the right side of diagram (1) and the left side of diagram (2). When a term appears as a necessary condition in one diagram and a sufficient condition in another, we can chain them together:

Harvard Law Took the LSAT Done a RC section

And we can also infer the following:

Harvard Law Done a RC section

When I say infer, I mean no one has told us that "If I go to Harvard Law, then I would have done a RC section". But because of the other information in diagrams (1) and (2), we do not need anyone to tell us this as we can simply infer this. The same interpretations apply: going to Harvard Law is sufficient to conclude that an RC section was completed, and completing an RC section is necessary for going to Harvard Law.

If A B and B C, then A B C, and also A C.

Now let's look at a case where chaining is not possible:

If I go to Harvard Law, then I would have taken the LSAT. I would have done a logical reasoning section only if I had taken the LSAT.

Diagramming:

Harvard Law Took the LSAT (1)

Done an LR section Took the LSAT (2)

Can we combine these? No. In both diagrams, "Took the LSAT" appears on the right side only. There is no arrow leading out from it, so there is nothing to follow. We cannot chain these two diagrams.

One more example before we move on. Try diagramming this yourself first:

If I go to Harvard Law then I would have taken the LSAT. I would not have taken the LSAT if I went to Harvard Medical School.

Diagramming:

Harvard Law Took the LSAT (1)

Harvard Medical Did NOT take the LSAT (2)

Can we combine these? At first glance it seems like no, since "Took the LSAT" doesn't directly match anything. But here is where contrapositives come in. The contrapositive of diagram (2) is:

Took the LSAT Did NOT go to Harvard Medical (3)

Now we can chain. "Took the LSAT" appears on the right side of diagram (1) and the left side of diagram (3):

Harvard Law Took the LSAT Did NOT go to Harvard Medical

And from this we can infer:

Harvard Law Did NOT go to Harvard Medical

Using contrapositives to enable chaining is a skill that comes up frequently, especially in must be true questions.

Unless

The keyword "unless" is the trickiest one to diagram. Honestly, I just memorized a heuristic for it rather than trying to derive the logic each time. Here is the trick:

Whatever comes after "unless" goes on the right side (necessary condition). The negation of everything else goes on the left side (sufficient condition).

Example:

Unless I am built different, I would have completed an undergrad degree before going to law school.

Applying the trick: "I am built different" goes on the right. The negation of the rest goes on the left:

Did NOT complete an undergrad degree before going to law school I am built different

The general rule: A unless B becomes not A B. This also holds when the structure is flipped: unless B, A also becomes not A B. Whatever is attached to "unless" goes on the right, and the negation of the remainder goes on the left.

Try this one on your own before reading the answer:

I would not be going to Harvard Law unless I did well on the LSAT.

Click to reveal

The answer:

I would be going to Harvard Law Did well on the LSAT

Note: the negation of "I would NOT be going to Harvard Law" is "I would be going to Harvard Law." Two negatives cancel out.

Take a moment to reflect on why this heuristic makes sense intuitively. It usually clicks after sitting with it for a bit, and having it internalized will save you real time on test day.

Some and Most

Everything we have covered so far deals with absolute conditional statements. Our running example:

If I go to Harvard Law, then I would have taken the LSAT.

is logically identical to:

All students who go to Harvard Law have taken the LSAT.

Both produce the same diagram. But what about a statement like:

Most students who go to Harvard Law have taken the LSAT.

(Harvard Law now accepts the GRE as an alternative to the LSAT, so this is actually a more accurate statement these days.)

We cannot diagram this as:

Harvard Law Took the LSAT

That diagram claims going to Harvard Law is sufficient to guarantee someone took the LSAT. But all we know is that most students did. Going to Harvard Law is no longer sufficient to make that inference.

So we need to change up our approach. First, let's start with the definitions:

Most means 50% + 1.

Some means at least 1.

If Harvard Law has 150 students, a "most" statement about taking the LSAT requires at least 76 students to have taken the LSAT.

On the other hand, a "some" statement about taking the GRE requires only 1 student to have taken the GRE. A "some" statement is satisfied even if every single student has that property — you just need to find one to prove it. For example, even if all 150 students took the GRE, I can still claim "some students at Harvard Law have taken the GRE".

Diagramming Most

Most students at Harvard Law have taken the LSAT.

We diagram this as:

Students at Harvard Law Took the LSAT

This arrow is one-directional, just like our conditional arrow. We cannot reverse it. Saying "most students who have taken the LSAT go to Harvard Law" would be a completely different claim (and a nonsensical one at that).

Chaining works the same way:

Most students at Harvard Law have taken the LSAT. Every student who has taken the LSAT has done an RC section.

Harvard Law Took the LSAT (1)

Took the LSAT Done an RC section (2)

"Took the LSAT" appears on the right of the first diagram and the left of the second. We can chain:

Harvard Law Took the LSAT Done an RC section

And directly:

Harvard Law Done an RC section

I hope this makes intuitive sense: if most students took the LSAT, and every LSAT taker did an RC section, then most students at Harvard Law did an RC section.

Also, the trigger words for a most diagram does not necessaarily have to contain "most". Other keywords and keyphrases include:

Mostly, Majority, Usually, More likely than not, More often that not.

Diagramming Some

Some students at Harvard Law have taken the GRE.

We diagram this as:

Students at Harvard Law Took the GRE

The key difference: the arrow points both ways. This means we can make inferences in either direction. We can say "some students at Harvard Law have taken the GRE" and also "some students who have taken the GRE go to Harvard Law."

Why does the reversal work? Because the burden of proof for a "some" statement is so low. If at least one Harvard Law student took the GRE, then at least one GRE taker goes to Harvard Law. The same person satisfies both claims.

And once again, the trigger words for a some diagram does not necessarily have to contain "some". Other keywords and keyphrases include:

Many, Several, Few, Sometimes, Little, Not all

A moment to spend on not all as many students find this a little trippy. Consider the following:

Not all law students are poli sci majors.

Even though we have all, the not part makes this a some statement. This is essentially saying "Some law students are not poli sci majors". So diagramming this gives us:

Law Students Not Poli Sci Major

Valid Inferences Table

Here is a summary of valid inferences when combining some and most statements:

Given	What can be inferred
A B and B C	A C
Most HL students took the LSAT. Every LSAT taker did an RC section.	Most HL students did an RC section.
A B and B C	A C
Some HL students took the GRE. Every GRE taker does a quant section.	Some HL students did a quant section.
A B and A C	B C

Let’s understand this last one a bit more since its a bit complicated. Assume Harvard Law has 150 students again.

When I say most students at Harvard Law have taken the LSAT, that means at least 76 students have taken the LSAT. When I say most students at Harvard Law have good GPAs, that means at least 76 students have good GPAs. But since the class size of Harvard Law is 150 exactly, there must be atleast one student who shares both properties of having a good GPA and have taken the LSAT (since 76+76 = 152). And since atleast one student would share both properties, I am allowed to write: Some students who have taken the LSAT have good GPAs!

Where No Valid Inference Can Be Made

Just as A B and C B cannot be chained, the following combinations also yield no valid inference:

A B, B C
A B, B C
A B, B C
A B, B C

Why? I will leave those as an exercise to the reader. (I always wanted to write that.) Seriously though, work through them using the Harvard Law 150-student example and see why no further inference is possible in each case.

How to Negate Correctly

Negation is central to contrapositives, so it is worth understanding how to do it properly. A negation is not simply the opposite of a phrase. It is the minimum thing you need to show in order to invalidate a statement.

Simple statements

The easiest ones. You just add or remove a "not."

Original: I like ice cream. (The most overrated dessert in existence, but that is beside the point.)

Negation: I do not like ice cream.

If a statement already contains a negation, removing it is your negation (because two NOTs cancel each other out).

Original: I do NOT think he is cool.

Negation: I think he is cool.

Universal statements (ALL)

To disprove an "all" statement, you only need to find one exception.

Original: All pigs fly.

Negation: Some pigs do not fly.

Existential statements (SOME)

To disprove a "some" statement, you would need to confirm there are zero exceptions — not a single one satisfies the property we are talking about.

Original: Some humans have wings.

Negation: All humans do not have wings. (yes even including those drinking Red Bull)

AND statements

To disprove an AND statement, you only need to disprove one of its parts. If I claim A AND B are both true, showing that even one of them is false is enough to invalidate the whole thing.

Original: I study consistently AND I study smart.

Negation: I do not study consistently OR I do not study smart.

Notice how the negation of AND becomes OR. You are saying: at least one of these does not hold.

OR statements

To disprove an OR statement, you need to disprove all of its parts. If I claim A OR B is true, showing that just one is false is not enough — you need to show both are false.

Original: I study consistently OR I study smart.

Negation: I do not study consistently AND I do not study smart.

Notice how the negation of OR becomes AND. You need both to fail for the OR statement to be disproved.

Here is a quick reference table for all of the above and more:

Original	Negation
I like ice cream.	I do not like ice cream.
I do not think he is cool.	I think he is cool.
All pigs fly.	Some pigs do not fly.
Some humans have wings.	All humans do not have wings.
I study consistently AND I study smart.	I do not study consistently OR I do not study smart.
I study consistently OR I study smart.	I do not study consistently AND I do not study smart.
I must go to the park.	It is not necessary for me to go to the park.
I cannot do this.	I could do this.

Pay special attention to last two as I did not cover those in detail but they should be quite self explanatory.

Contrapositives with AND / OR

You already know the basic contrapositive rule:

A B becomes Not B Not A

But when AND or OR appears in a diagram, you need to negate the entire side correctly before flipping. Remember: the negation of AND is OR, and the negation of OR is AND.

Contrapositive with AND in the necessary condition

Original: Go to Harvard Law Good GPA AND Good LSAT score

Contrapositive: Not a good GPA OR Not a good LSAT score Did not go to Harvard Law

Negating the necessary condition (AND becomes OR) and flipping gives us: if either condition fails, we can conclude the person did not go to Harvard Law.

Contrapositive with OR in the sufficient condition

Original: Consistent studying OR Smart studying Good score

Contrapositive: Not a good score Not consistent studying AND Not smart studying

Negating the sufficient condition (OR becomes AND) and flipping gives us: if you did not get a good score, then neither condition was met.

Conclusion

That is the end of the foundational skills chapters. Starting with the next chapter, we are finally solving real LSAT questions.

I may have slightly undersold this chapter at the start. If you feel overwhelmed right now, that is completely okay. To help you lock this in, I have put together three problem sets covering everything in this chapter. Work through them like this: do a problem set, identify where you struggled, review those sections, and then do the next problem set with a fresh set of questions.

You do not need to complete all three. Once you can get through a problem set reasonably well, move on. The next chapter is where the fun actually begins as we will be finally solving real LSAT questions.

Also, the next few topics do not require diagramming, so take a breather if you need one. Diagramming comes back in full force for Sufficient Assumption questions which is towards the end of Chapter 2, and again in Chapters 3 and 5. The point is that the time spent on these fundamentals is not wasted. When you need it, the skill will already be there.