Free LSAT Course > Logical Reasoning > Formal Logic > Necessary And Sufficient Conditions

Knowing Formal Logic is necessary for taking the LSAT. This content may be alien to you, so we start slowly and move through the topics in a simple and intuitive fashion.

The most basic logic concept is necessary vs. sufficient.

Necessary or Sufficient?

Sufficient

“P is sufficient for Q” means:

The truth of P is enough for the truth of Q
P cannot be true without Q being true
If P is true, then Q is true

(informally) P is all you need for Q

Necessary

“P is necessary for Q” means:

Q cannot be true without P being true
Q is not true unless P is true
If Q is true, then P is true
Q is true only if P is true

(informally) P is needed for Q, can’t have Q without P

Necessary and Sufficient Conditions Part 1 Video Summary

  • 00:15 – The premises of a causal argument provide support for the hypothesis that one event causes another.
  • 1:08 – Cause is often used to describe something that has a specific outcome as a result.
  • 2:12 – A necessary cause is an essential condition. It must be present for its corresponding conclusion to be present.
  • 2:28 – A sufficient cause will always be enough to produce the effect. If it is present, then its corresponding conclusion must also be present.
  • 5:58 – There are four possible combinations of necessary and sufficient conditions: necessary but not sufficient, sufficient but not necessary, both sufficient and necessary, and neither sufficient nor necessary.
  • 6:52 – If P is necessary for Q, then Q cannot be true unless P is true. Q is true only if P is true. This can be diagrammed as follows: Q → P.
  • 7:36 – If P is sufficient for Q, then P’s being true is enough to make Q true. If P is true, then Q is true. This can be diagrammed as follows: P → Q.

Being in Italy is sufficient for being in Europe.
Being in Europe is necessary for being in Italy.

Being in Italy is not necessary for being in Europe (one can still be in Europe outside Italy, in Greece for example).
Being in Europe is not sufficient for being in Italy (for the same reason as above).

Necessary and Sufficient Conditions Part 2 Video Summary

  • 00:26 – To say that P is a necessary condition for Q is to say that “Q is true only if P is true”. This can be diagrammed as follows: Q → P.
  • 00:44 – To say that R is a sufficient condition for S is to say that “If R is true, then S is true”. This can be diagrammed as follows: R → S.
  • 1:01 – There are four possible combinations of necessity and sufficiency: necessary and sufficient, necessary but not sufficient, sufficient but not necessary, and neither necessary nor sufficient.

Next LSAT: September 16

Necessary, Sufficient, Both or Neither?

Identify whether each of the following statements presents a necessary condition, a sufficient condition, both, or neither.

  • P is a necessary but not sufficient condition for Q: Q cannot be true without P being true,
    but P is not enough for Q.
  • P is a sufficient but not necessary condition for Q: P is enough for Q, but Q can be true
    without P being true.
  • P is a condition that is both sufficient and necessary for Q: P is enough for Q, and Q cannot be
    true without P being true.
  • P is neither a sufficient nor necessary condition for Q: P is not enough for Q and Q can be
    true without P being true.

Do not make any extraordinary assumptions when answering these questions.
(Remember the notion of background assumptions from the above video at 1:52)

Watering a plant is _______________ for it to grow.

Explanation

Necessary but not Sufficient

Watering a plant is a necessary but not sufficient condition for it to grow. It is necessary because water is required for plants to grow. However, it is not sufficient because plants also need other things for growth, such as sunlight and proper soil.

Finishing college is _______________ for one to be rich.

Explanation

Neither Necessary nor Sufficient

Finishing college is neither necessary nor sufficient for being rich. Some people did not finish college but still became rich. Moreover, finishing college does not guarantee that a person will become rich.

Knowing the correct formula is  _______________ for finding the volume of an object in a math exam.

Explanation

Necessary but not Sufficient

Knowing the correct formula is a necessary condition for solving the volume of an object in a math exam because using an incorrect formula will lead you to a wrong answer. However, it is not a sufficient condition because knowing the formula is not enough. You still have to do the calculation correctly to get the correct volume.

Given that the passing score for the midterm science exam is 70%, getting a score of 70% or higher in the exam is _______________ for passing it.

Explanation

Necessary and Sufficient

It is both necessary and sufficient that one gets a score of 70% or higher in the midterm science exam in order to pass it. It is a necessary condition because the passing score for the midterm science exam is precisely 70%, so getting 70% or higher is required for you to pass it. Moreover, it is also a sufficient condition because getting a passing score automatically means that you pass the exam.

Living in Hong Kong is _______________ for living in Asia.

Explanation

Sufficient but not Necessary

Living in Hong Kong is a sufficient but not necessary condition for living in Asia. If one lives in Hong Kong, then it automatically means that they also live in Asia, since Hong Kong is in Asia. However, it is not necessary because one can live in other countries in Asia other than Hong Kong.

Every working day, John takes the metro line D at 8:30, which takes him to work in 35 minutes. His work regularly starts at 9:30. However, tomorrow his shift starts at 9:00. Therefore, it is _____________________ condition that he takes the same line at least five minutes earlier than usual for not being late to his work tomorrow.

Explanation

Both Necessary and Sufficient

It takes John 35 min to get to work. It is enough to take the metro 5 min earlier than the usual 8:30 to arrive at 9:00. Therefore, the condition is sufficient. If he fails to depart at least 5 min earlier, he will have less than 35 min to travel and will be late. He needs to depart at least 5 min earlier. Therefore, the condition is necessary. (Note that taking the metro exactly 5 min earlier is not necessary for being on time, because he can depart 15 min earlier and not be late; however, our condition is not to depart exactly, but at least 5 min earlier, which is necessary for arriving at 9:00.)
Therefore, the condition is both sufficient and necessary.

Necessary or Sufficient?

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Next LSAT: September 16