Some, All, Most on the LSAT

So far we’ve dealt with if then statements that are categorical: No Great White Sharks are vegetarians.

Now we’re adding a level of complexity with some, all, and most. We’ll eventually visualize these relationships with Venn Diagrams.

This first video is beginner level and goes through the formal logic meanings of the following:

  • Most
  • Some
  • Unless
  • Only If
  • Or

Five Simple But Dangerous LSAT Vocabulary Terms

  • 0:10 – There are 5 seemingly simple words or phrases that many LSAT takers understand incorrectly. These are: most, some, only if, unless¸ and or.
  • 1:50 – The word “most” has two slightly different meanings based on context. In some cases, it simply means more than half. While in other cases, it implies that it means more than half but not all.
  • 2:01 – On the LSAT, “most” takes on the more general meaning which is “more than half” and does not excluded the possibility of all.
  • 3:19 – The word “some” also has two different meanings. In some cases, it takes on a more general meaning which is “an unknown amount”. While sometimes it is more specific which means “an unknown amount that is a minority”.
  • 3:35 – On the LSAT, “some” takes on the more general meaning which is “an unknown amount” which has to be greater than zero and up to 100%.
  • 4:33 – The phrase “only if” also has two different meanings. The first meaning is that in a statement, if we have the thing after the phrase “only if”, then it guarantees the thing before this phrase. The second meaning is that in a statement, if we have the thing before the phrase “only if”, then the thing after it is guaranteed.
  • 5:30 – On the LSAT, “only if” matches the second meaning we presented. The thing before the phrase “only if” guarantees the thing after it.
  • 6:33 – The phrase “unless” also has two different meanings. The first meaning implies that the thing after “unless” guarantees the negation of the thing before it. The second meaning is quite the opposite. It implies that the negation of the thing before “unless” guarantees the thing after it.
  • 7:10 – On the LSAT, “unless” is used similarly as the second meaning we presented. It implies that the negation of the phrase before the words “unless” guarantees the phrase after it.
  • 8:00 – The word “or” can be interpreted in two ways. First, it can imply the possibility of both. On the other hand, its second meaning does not allow the possibility of both.
  • 8:11 – On the LSAT, the “or” is used as inclusive, which means that there is the possibility of both, unless stated otherwise.
  • 8:22 – Recap:most” means more than half; “some” means an unknown amount greater than zero; A “only if” B means that A guarantees B; not A “unless” B means that A guarantees B; and “or” is inclusive, unless stated otherwise.

Basics of Venn Diagrams

One way to visualize most, some, and or is to diagram them using overlapping circles (Venn Diagrams).

Venn Diagrams Video Summary

  • 0:11 – Categorical propositions are statements that claim a “Subject category” has a characteristic identified by the “Predicate category”.
  • 0:29 – For example: Dolphins are mammals. Here, the category “Dolphins” is being related to the category of “mammals”. It says that all of the Dolphin category is also part of the mammal category.
  • 0:42 – There are four standard propositional forms in categorical (classical logic) called “claims“.
  • 1:00 – These claims are: All S are P. (A claim); No S are P. (E claim); Some S are P. (I claim); Some S are not P. (O claim).
  • 1:11 – For example: Bicycles have two wheels. This is an example of A claim. It claims that all the type of things called “bicycles” are also the type of thing that has two wheels.
  • Other examples: No bicycles are things that fly. (E claim); Some bicycles are red. (I claim); Some bicycles are not things with a bell. (O claim).
  • 1:35 – Venn Diagrams are a way to visualize the relationship between a Subject and a Predicate in categorical propositions. Two overlapping circles are used to show this relationship regardless of the form of the proposition. Shading and an “x” are used to show these relationships.
  • 2:14 – In a Venn Diagram, a category where there are no examples is shaded. The diagrams of claims are the same for each category, no matter the content.
  • 2:51 – An A claim says something about all of the Subject, but it does not say anything about all of the Predicate. Its Venn Diagram has the left side of the circle of the Subject shaded.
  • 3:16 – In the Venn Diagram of an E claim, the intersection of the two circles is shaded. It says that there are no examples of the Subject that is also a part of the Predicate.
  • 3:33 – Venn Diagrams also make us of an “x” to indicate “some” or “some are not”. “Some” in categorical logic means “at least one”.
  • 3:53 – In the Venn Diagram of an I claim, the “x” is in the area where the two circles overlap. This means that there is at least one thing that is both the Subject and the Predicate.
  • 4:11 – In the Venn Diagram of an O claim, the “x” is inside the Subject circle and outside the Predicate circle. This indicates that there is at least one Subject that is not part of the Predicate.

Venn Diagram Examples

At Beverly Hills University (BHU), no students are on financial aid (FA).

 

We can also understand the given statement as: if you are a student at Beverly Hills University, then you are not on financial aid. This means that the BHU and FA circles should not have any intersecting areas.

All grad students (GS) are higher-ed students (HES).
Some grad students (GS) are on financial aid (FA).

 

Given that all grad students are higher-ed students, the whole GS circle must be inside the HES circle. Moreover, since some grad students are on financial aid, then the GS and FA circles must have intersecting areas.

Next LSAT: January 13th

Premise 1: Most voters view Hillary unfavorably.
Premise 2: Most voters view Trump unfavorably.

Inference?

Since some voters view both Hillary Clinton and Donald Trump unfavorably, the circle representing the voters who view Hillary Clinton unfavorably should intersect with the circle representing the voters who view Donald Trump unfavorably.

The area with “x” represents the voters who view both Hillary Clinton and Donald Trump unfavorably, which is the inference than can be drawn.

Choose the diagram that best represents the given statement.

All national scientists (S) are Ph.D. graduates (P), but not all Ph.D. graduates (P) are national scientists (S).

Explanation

Diagram 2: The statement claims that if you are a national scientist (S), then you have to be a Ph.D. graduate (P). The “X” should then be placed where it is both within the (S) and (P) circles. Both the first and second diagrams meet this criterion. However, the (S) circle should also be inside the (P) circle because the statement tells us that all (S) are (P). Therefore, the second diagram is the correct answer.

Diagram 1: Although this diagram meets the criterion that “X” is both within the (S) and (P) circles, the whole (S) circle should be inside the (P) circle because the statement tells us that all (S) are (P). However, this diagram shows that some (S) are not (P), which is incorrect.

Diagram 3: This diagram does not show what the statement represents. The “X” should be placed where it is both within the (S) and (P) circles, but this diagram only shows “X” to be within the (S) circle and not the (P) circle. Therefore, this is not the correct answer.

Some animals (A) are not reptiles (R), but all reptiles (R) are animals (A).

Explanation

Diagram 2: The statement claims that there are animals that are not reptiles so the circle of animals (A) should have a part that does not intersect the circle of reptiles (R). Both the first and second diagrams meet this criterion, but the first one implies that there are reptiles that are not animals, which is wrong. The second best represents the statement that not all animals are reptiles because the whole (R) is inside (A), while not all (A) intersect (R).

Diagram 1: Although this diagram meets the criteria that circle (A) should have a part that does not intersect circle (R), it shows that there are (R) that are not part of (A). Therefore, it is incorrect.

Diagram 3: This diagram does not show any intersection between circle (A) and circle (R), but the statement shows that all (R) are (A) and thus, should have an intersection. This diagram presents that there is no intersection between (A) and (R), which is incorrect.

Some of the members of the organization (M) are foreigners (F).

Explanation

Diagram 1: The statement tells us some members of the organization (M) are foreigners (F). This means that there should be an intersection between the circle (M) and the circle (F). The first diagram shows exactly this relation, and it also shows that there are parts of (M) that are not part of (F) and there are parts of (F) that are not part of (M).

Diagram 2: This diagram shows the circle (M) and circle (F) are overlapping, which means that all (M) are (F), and all (F) are (M). However, the statement tells us that only some (M) are (F). Therefore, this is incorrect.

Diagram 3: This diagram shows that circle (M) and circle (F) have no intersection, but the statement claims that some (M) are (F), and thus, should have an intersection. Therefore, this is incorrect.

Choose the diagram where the “x” matches the “some” statement and where the shade is based on the "all" statement.

All businessmen (B) love money (M).
Some politicians (P) are businessmen (B).

Explanation

Diagram 1: The first statement claims that all (B) are (M), which means that parts of circle (B) that do not intersect circle (M) should be shaded. Moreover, the second statement states that some (P) are (B), which means that the “X” should be within the intersection of (P) and (B) that is not shaded. Hence, this is the correct answer.

Diagram 2: Although this diagram correctly presents the second statement, it is incorrect regarding the first statement. The shaded part should be where (B) does not intersect (M), but this diagram shows it the other way around. The parts of (M) that do not intersect (B) is shaded, which is the opposite.

Diagram 3: This diagram is simply incorrect because the shaded part does not represent the statements given in the question. The first statement claims that all (B) are (M), so the parts of circle (B) that do not intersect circle (M) should be the ones shaded and not parts of (P).

All who passed the Professional Exam (P) also passed the “logical reasoning” section of the exam (L).
Some people who passed the “reading comprehension” section (R) also passed the Professional Exam (P).

Explanation

Diagram 2: The first statement presents that all (P) are (L) and thus, any (P) that does not intersect with (L) must be shaded. Furthermore, the second statement presents the intersection between circle (R) and circle (P). This means that “X” should be within the intersection of (R) and (P) that is not shaded.

Diagram 1: This diagram shades both the whole of circle (L) and circle (R). This is incorrect because it leaves only circle (P) remaining, and it presents relations that are far from what the statements represent.

Diagram 3: Although this diagram correctly shows the intersection of (R) and (P), the first statement presents that any (P) that does not intersect with (L) must be shaded. However, this diagram shows that (L) not intersecting with (P) is shaded, which is different from the meaning of the first statement. Therefore, this diagram is incorrect.

All of my college friends (F) will go to the party (P).
Some of my classmates in college (C) are also my friends (F).

Explanation

Diagram 2: The first statement presents that all (F) are (P). This means that parts of (F) that do not intersect with (P) should be shaded. Moreover, the second statement shows the intersection of (C) and (F). Therefore, the “X” should be within the part of the intersection of (C) and (F) that is not shaded.

Diagram 1: This diagram shows the whole circle (C) shaded, but the first statement states that all (F) are (P). This means that parts of (F) that intersect with (P) should not be shaded. Therefore, this diagram is incorrect.

Diagram 3: This diagram correctly shows the intersection of (C) and (F) as stated in the second statement. However, it incorrectly represents the first statement. The first statement shows that parts (F) that do not intersect with (P) should be shaded, but this diagram shades the intersection of (F) and (P), which makes this diagram incorrect.

Negations

What if you take a contrapositive of the descriptive statements above? How do you describe the negation of all? The following video explains the logical significance of these terms:

  • Some are not
  • Most are not
  • None

Logical Quantifiers “all” And “some are not” Video Summary

  • 0:33 – Consider the following argument: All people are mammals, and some people are lawyers. No sharks are mammals. Therefore, no lawyers are sharks.
  • 0:53 – This is an example of a formal logic argument which has a specific vocabulary and structure.
  • 1:10 – The six most important qualifiers in formal logic arguments are: all, most, some, some are not, most are not, and none.
  • 1:32 – Note: These questions will require deductions from given information, which requires an understanding of qualifiers and how to represent information.
  • 2:06 – For example: All dogs are animals.
  • 2:12 – “All” means everyone or everything, which also means that every element in the first group is part of the second group. In the percent bar below, we can see that “all” corresponds to 100%.
  • 2:33 – The statement “All dogs are animals” simply says that every dog is also an animal. This is similar to saying that “100% of dogs are animals”.
  • 3:12 – The negation of this statement is: Some dogs aren’t animals. Note: The negative of “all” is “some are not” and vice versa. Its negative is not “none”.
  • 4:01 – Another example: Some animals aren’t pets. Some are not” means that at least one isn’t and possibly none are. There is at least one element in the first group that isn’t part of the second.
  • 5:07 – In the percent bar below, we can see that “some are not” corresponds to everything under 100% up to and including zero percent.
  • 5:40 – Returning to our original example: All dogs are animals. This statement is not reversible. This means that if a statement says that all A’s are B, we can’t automatically assume that all B’s are A.
  • 6:23 – Similarly, the statement “Some animals aren’t pets” is not reversible. If a statement says that some A’s are not B, we can’t automatically assume that some B’s are not A.
  • 7:06 – Qualifiers are best presented visually. (Refer to the modality with Venn Diagrams in the sub-section below.)
  • 10:26 – Recap: All” means 100%, while “some are not” means anything less than 100%. They’re each other’s negatives, and they are both not reversible. Regarding diagrams, for “all”, the first circle has to be completely inside the second. For “some are not”, the first circle has to have a part of it outside the second.

You would think the opposite of all would be none, but it isn’t. The opposite of all (which you are going to be using often on the LSAT if you are taking the contrapositive of an all statement) is some are not.

Negation of all, some, and some are not.


All
means 100% of something.

Some means greater than 0% (none) but not 100% (all).

Some are not means below 100% (all) down to 0% (none).

Converse of all and some are not.

All dogs (D) are nice animals (NA).

Is all nice animals are dogs a valid inference? What is the mistake?

Explanation

The converse of “all” statements isn’t valid.

For example:
All dogs (D) are nice animals (NA).

All D → NA is not logically equivalent to NA → D (All nice animals are dogs).

NA → D is invalid.

All dogs (D) are nice animals (NA).

What is the negation of that statement? Is it a valid inference?

The negation of “all” is “some are not“.

The negation of “All dogs are nice animals” is not No dogs are nice animals”. Its negations is “Some dogs are not nice animals”.

Note: The opposite of “nothing” is “some“.

Negations aren’t valid inferences.

“Life is a great adventure or nothing”
-Helen Keller

Explanation

The contrapositive (show in red above) shows that the opposite of nothing is something (the opposite of nothing isn’t everything).

Examples: All A are B

Evaluate whether the following inferences are valid or invalid.

All LSATs (L) are hard tests (HT).

All L → HT

1. Some tests that are not hard are not an LSAT.

Explanation

Some ~HT → ~L

This is a valid inference because this is precisely the contrapositive of the original statement. A statement and its contrapositive are logically equivallent.

2. All hard tests are LSAT tests.

Explanation

ALL HT → L

This is an invalid inference because this is the converse of the original statement. A converse is not equivalent to the original statement.

3. Some LSATs are not hard.

Explanation

Some L → ~HT

This is an invalid inference because it directly contradicts the original statement.

All licensed engineers (LE) passed the professional engineering exam (P).

All LE → P

1. All people who passed the professional engineering exam are licensed engineers.

Explanation

All P → LE

This is an invalid inference because this is the converse of the original statement. A statement and its converse are not logically equivalent.

2. Some people who did not pass the professional engineering exam are not licensed engineers.

Explanation

Some ~P → ~LE

This is a valid inference because it is the contrapositive of the original statement. The contrapositive of a statement is logically equivalent to the statement itself.

3. Some licensed engineers did not pass the professional engineering exam.

Explanation

Some LE → ~P

This is an invalid inference because it directly contradicts the original statement.

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Official LSAT Questions

This can be a confusing question, but all it’s really asking you to do is use substitution (athletes for bankers).

The three dots symbolize the conclusion.

Video Summary

  • 0:08 – Consider this example: At a gathering at which bankers, athletes, and lawyers are present, all of the bankers are athletes and none of the lawyers are bankers. The question is: If the statements above are true, which one of the following statements must also be true?
  • 0:22 – Its better to solve this type of questions via diagram.
  • 0:28 – What we know: All of the bankers are athletes. Diagram this as B → A; None of the lawyers are bankers. Diagram this as: L → ~B.
  • 0:40 – Take the contrapositive of the second statement: If you’re a banker then you’re not a lawyer. Diagram this as: B → ~L.
  • 0:54 – Given that if you’re a banker then you’re an athlete and if you’re a banker then you’re not a lawyer, we can conclude that some people who are athletes are not lawyers. Look for this in the answer choices and that’s our correct answer.

LSAT Quiz

Some, All, Most

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Next LSAT: January 13th