Free LSAT Course > Logical Reasoning > Formal Logic > Formal Logic Questions

## Chapter Summary

### I. LSAT Formal Logic

LSAT Formal Logic

The left part of a conditional statement (If . . . then . . .) is called the sufficient condition, while the right part is called the necessary condition.

We diagram conditional statements by combining the sufficient and necessary conditions using an arrow ().

sufficient condition → necessary condition

A necessary cause is something that is essential to produce an effect. A sufficient cause is something that is enough by itself to produce an effect.

There are four possible combinations of necessary and sufficient conditions:

• necessary but not sufficient
• sufficient but not necessary
• both sufficient and necessary
• neither sufficient nor necessary

### II. DIAGRAMMING CONDITIONALS

Diagram Conditionals

The first step in diagramming a conditional is to look for the keyword that will tell you which phrase is the necessary condition and which one is the sufficient condition.

Sufficient indicators:

• If, If only
• All, Any, Each, Every
• When, Whenever, Whoever, Whatever
• People who, In order

Necessary indicators:

• Only, Only if, Only when
• Relies on, Depends on
• Must, Requires

### Steps for diagramming conditionals

1. Locate the conditional indicator (keyword) in the statement.
2. Locate the conditional phrase after the conditional indicator.
3. Abbreviate the conditional phrase.
4. Locate the conditional result in the conditional statement. (This is the phrase other than the conditional phrase.)
5. Abbreviate the conditional result.
6. Finally, combine the two conditional phrases in the form: sufficient phrase → necessary phrase.

### The negation symbol/tilde symbol (~)

1. We use (~) when the sufficient indicator is “no/none” or “never.” Using these means we also have to negate the necessary condition using (~).
2. The necessary indicators “unless,” “until,” and “except” also require the negation of the sufficient condition using (~).

### III. Contrapositive

Contrapositive

You get the contrapositive of a conditional statement when you negate both the sufficient and necessary conditions and switch their positions. When converting a conditional statement to its contrapositive, its sufficient condition is negated and becomes the necessary condition, while its necessary condition is negated and becomes the sufficient condition.

• Conditional statement: sufficient phrase → necessary phrase
• Contrapositive: ~necessary phrase → ~sufficient phrase

The idea of a contrapositive is important because it is one of the few statements that you can validly infer from a conditional statement.

### IV. At Least One

At Least One

The statement If not A → B means that at least one of A and B is true. The only criterion is that at least one of these is true, which means that both can also be true at the same time.

The combination of a conditional statement and its contrapositive form the statement “If not A → B.”

1. Premise 1: ~A → B
2. Premise 2 (the contrapositive of premise 1): ~B → A
3. Conclusion: (~A → B) and (~B → A), or If not A → B

### V. If and Only If

If And Only If

If and only if” statements, or biconditional statements, are statements composed of two conditional statements namely, a conditional statement and its converse. Biconditional statements use the left-right arrow (↔).

You can get the converse of a statement simply by switching the position of its conditions (e.g., conditional statement: A → B; converse: B → A).

From the biconditional A ↔ B, we can infer the following statements:

• A → B
• B → A
• (A → B) and (B → A)

Inverse
The inverse of a conditional statement is simply a conditional statement where both of its conditional phrases are negated.

• A → B
• Inverse: ~A → ~B

The inverse can also be understood as the contrapositive of the converse and vice versa.

• A → B
• Converse: B → A
• Contrapositive of the converse: ~A → ~B, which is the same as the inverse (~A → ~B)

### VI. Conditional Conjunctions

Conditional Conjunctions

Transitive Property
The transitive property comes in the following form:
If A → B and B → C, then A → C

“And” Conditionals
“And” Conditionals are conditional statements where one of the conditions is a compound condition (more than one element) with the conjunction “and.

An “and” conditional comes in the following form:
If A and B, then C or (A and B) → C

The contrapositive of an “and” conditional follows the same principles of contraposition (negate both the sufficient and necessary conditions and switch their positions).

The diagrams of an “and” conditional and its contrapositive are as follows:

• (A and B) → C
• Contrapositive: ~C → ~ (A and B), which can be further simplified to ~C → (~A or ~B)

The same principles apply when the compound condition is on the other side of the conditional:

• A → (B and C)
• Contrapositive: ~ (B and C) → ~A, which can be further simplified to (~B or ~C) → ~A

“Or” Conditionals
“Or” Conditionals are conditional statements where one of the conditions is a compound condition (has more than one element) with the conjunction “or.

An “or” conditional comes in the following form:
If A or B, then C or (A or B) → C

The diagrams of an “or” conditional and its contrapositive are as follows:

• (A or B) → C
• Contrapositive: ~C → ~ (A or B), which can be further simplified to ~C → (~A and ~B)

The same principles apply when the compound condition is on the other side of the conditional:

• A → (B or C)
• Contrapositive: ~ (B or C) → ~A, which can be further simplified to (~B and ~C) → ~A

### VII. Invalid Inferences

Invalid Inferences

An invalid inference occurs when you incorrectly infer a statement from another conditional statement. Invalid inferences are only concerned with the form of conditional statements and not their content.

Fallacy of the Converse
The fallacy of the converse is incorrectly inferring that a conditional statement is logically equivalent to its converse. You cannot infer the converse of a conditional statement from itself because they are logically different statements.

A → B is not logically equivalent to B → A

Fallacy of the Inverse
The fallacy of the inverse is very similar to the fallacy of the converse. The difference is that the fallacy of inverse is incorrectly inferring that a conditional statement is logically equivalent to its inverse.

A → B is not logically equivalent to ~A → ~B

Moreover, the fallacy of the inverse and the fallacy of the converse are ultimately the same because they are the contraposition of the other.

To better understand the relations between a conditional statement and its contrapositive, converse, and inverse, you can use the diagram below as reference: ### VIII. Some, All, Most

Some, All, Most

Understanding the “some” and “some are not” qualifiers and how to represent them is important for passing the LSAT logic section.

• All means 100%.
• Some” means greater than 0% but less than 100% (1% to 99%).
• Some are not” means less than 100%, but unlike “some,” it also includes 0%.

Note: The opposite/negation of “all” is NOTnone.” The opposite/negation of “all” is “some are not.” Remember not to confuse these because a lot of people make this common mistake, which can easily be avoided by always paying close attention. These qualifiers are not reversible.

### IX. Must Be True Questions

Must Be True Questions

Most LSAT logic questions deal with “must be true” or “must be false” questions. These questions require that conclusions are logically inferred or properly concluded from the premises. There is no space for probability or possibility in these scenarios, everything either “must be true” or “must be false.”

Must be true
The best way to solve “must be true” questions is to diagram them and identify their parts and relations.

This type of question in the LSAT will have a total of five choices. Four of the choices in a “must be true” question can be correct, which means that they can be wrong or are only likely but not guaranteed to be true. Only one of the answers must be correct, which means that it is always true given the information provided by the question and is the correct answer.

Must be false
Likewise, “must be false” questions can also be more easily solved by diagramming them and identifying their parts and relations.

Similar to “must be true” type of questions, this type of question will have a total of five choices. Four of these can be false or cannot be directly inferred from the question, but only one must be false, and this is the correct answer.

### X. Most Strongly Supported

Most Strongly Supported

Most strongly supported” questions in the LSAT are questions where you have to find the answer that is most strongly supported by a given set of premises.

If the question asks for something that is most supported by the given information but does not necessarily have to be true, then it is a “most strongly supported” question.

Steps for solving “most strongly supported” questions:

1. Look for the relation among the statements.
2. Analyze the logical structure of the set of premises and search for language cues to determine what kind of reasoning the question has.
4. Review the answer choices and determine which one best fits the given premises.

Tips for solving “most strongly supported” type of questions:

• As with most other logic questions, the best way to clarify and analyze a set of premises is to diagram them.
• Read and understand the question carefully and go through all the answer choices attentively.
• There are times when the correct answer is just a restatement of one of the premises, so it is useful to find valid inferences from the given set of premises.
• Remember that “most strongly supported” questions are different from “must be true” questions.
• Always be wary of invalid inferences and trap choices.

### Tricks to make LSAT questions more difficult video summary

• 2:30 – It is better for students to spend time figuring out what the wrong answers are. Eliminating the wrong answers to find the correct one is a more efficient strategy in the long run.
• 2:40 – Distractor strategies show up over and over again, which means that students will sometimes take too much time finding the right answer. This is why eliminating the wrong answers first is the best way to go.
• 3:21 – An example of a distractor strategy is presenting comparisons as absolutes or vice versa. This is common in many questions. Students only have to read the passages and answer choices carefully in order to not fall for these traps.
• 4:00 – These distractor strategies eventually become obvious once you know what to look for.

Next LSAT: Jun 10/Jun 11

#### Review Questions

Below are some practice questions for a review of the Formal Logic chapter.