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# Logical Reasoning Challenge Question: Inference #1

## The Question

All professors and physicians are doctors. Most qualified doctors have PhDs. But if a doctor is inexperienced then he or she is unqualified, and all experienced doctors are highly educated.

If all of the above statements are true, then which one of the following must be true?

(A) Any physician that is highly educated is experienced.
(B) Most doctors who have PhDs are qualified.
(C) Most professors who are qualified have PhDs.
(D) Any qualified professor or physician is highly educated.
(E) Any professor who is highly educated is a qualified doctor.

## The Prompt

The prompt tells us that all of the facts provided are true. Even if some of them seem unlikely to you, it doesn’t matter. For this question, we pretend that they are all perfectly accurate.

More importantly, we pretend like we know nothing else about doctors, physicians, or professors. The only information available to us is what’s provided in the stimulus. Our answer must be supportable using that–and only that–information.

So, let’s take a look at the information we’re given and organize it as clearly as possible.

## The Stimulus

First, we’re going to tackle the stimulus. Let’s assign some variables to the different characteristics of doctors that are mentioned in the text.

qualified = B
PhD = C
experienced = D
educated = E

We can now rewrite the stimulus as:

All professors and physicians are doctors. Most B doctors have Cs. But if a doctor is not D then he or she is not B, and all D doctors are E.

Let’s organize our facts more clearly by working through the stimulus one phrase at a time, writing out some if/then statements and contrapositives, if possible.

1. All professors and physicians are doctors
2. If/then form: If prof or phys → doctor
Contrapositive: if not doctor → not prof and not phys

[Notice how anything that’s true of all doctors will also be true of all physicians and professors. This is important because all of the remaining statements in the stimulus apply to all doctors, and therefore also to all physicians and professors.

Meanwhile, anything that’s true of some or all physicians and/or professors will also be true of at least some doctors. However, something that’s true of some or most doctors will not necessarily be true of any professors or physicians.]

The rest of the statements in the stimulus apply to all doctors.

3. Most qualified doctors have PhDs.
4. Most Bs have Cs.

5. If a doctor is inexperienced then he or she is unqualified.
6. If/then form: If not D → not B
Contrapositive: if B → D

7. All experienced doctors are highly educated.
8. If/then form: If D → E
Contrapositive: if not E → not D

Finally, let’s see if we can combine any rules and come up with some deductions.

• The 4th statement (if D → E) can be combined with the contrapositive of the 3rd statement (if B → D) to come up with: If B → D and E
• We can also combine the 3rd statement with the contrapositive of the 4th statement: if not E → not B.

That’s probably about all we can do at this point, so let’s move on to the answer choices.

(A) Any physician that is highly educated is experienced.

Using our assigned variables, we can translate this as, if E → D. In this statement, E (being highly educated) is the trigger. In other words, it says that if a physician is highly educated, then we definitely know that physician is also experienced.

Unfortunately, the stimulus doesn’t support this. The stimulus tells us that an experienced physician will definitely be highly educated, but not necessarily the other way around. In other words, E (being highly educated) isn’t a trigger in any of our factual statements above. We have not E as a trigger in two different places, but that’s the closest we get. Therefore, this answer choice is not supported by the facts provided.

(B) Most doctors who have PhDs are qualified.

Using our variables, we can translate this as, most Cs are Bs. Again, this is the opposite of what the stimulus tells us. We know that most Bs (qualified doctors) have Cs (PhDs), but we can’t say that the reverse is true.

Here’s an example that illustrates why this answer choice is wrong: we have 100 doctors total. Of the 100, 20 of them are qualified and 90 have PhDs. A total of 18 doctors are qualified and have PhDs. Thus, most of the qualified doctors have PhDs (18 out of 20). However, it isn’t accurate to say that most doctors with PhDs are qualified (only 18 out of 90).

(C) Most professors who are qualified have PhDs.

Using our variables, we can translate this as, most Bs have Cs. This is exactly the same as our statement #2 up above.

However, this statement is about professors, whereas the original was about doctors. Remember that in the diagram above we discovered that something that is true of all doctors is also true of all professors. Unfortunately, it isn’t true that all qualified doctors have PhDs. It’s true that most qualified doctors have PhDs.

Let’s draw out a scenario: We have 20 professors within a group of 100 doctors. 80 of our doctors are qualified and 41 of those 80 have PhDs. Therefore, most of our qualified doctors have PhDs. Is it necessarily true that most of our qualified professors have PhDs? Nope. What if only 10 of our professors are qualified, and none of those professors is among the 41 qualified doctors who have PhDs? That would look something like this:

Thus, we can see how it’s possible to have no qualified professors with PhDs, even within a group of doctors, in which most of the qualified ones have PhDs. Therefore, this answer choice isn’t necessarily true.

(D) Any qualified professor or physician is highly educated.

This is the correct answer choice.

Using our variables, we can translate the statement as, if B → E. Looking back at the deductions we made earlier, we discovered that you could combine the 4th statement (if D → E) with the contrapositive of the 3rd statement (if B → D). This gave us the statement, if B → D and E.

That statement is true of all doctors, which means it’s also true of all physicians and all professors. Therefore, this answer choice is fully supported by statements made in the stimulus.

(E) Any professor who is highly educated is a qualified doctor.

We can translate this answer choice as, if E → B. Again, this answer uses E as a trigger, and as we discovered in the first answer choice, we have no statement in the stimulus that uses E as a trigger. Therefore, this is an insupportable statement. We don’t even need to worry about comparing professors and doctors in this case.

## The End

I hope this problem was a productive challenge for you. If you have any questions about the right or wrong answers, the diagrams, or the formal logic, don’t hesitate to post a comment. Otherwise, check back soon for more Logical Reasoning Challenge Questions!

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