Here are some practice questions for a review of the Formal Logic chapter.
These are simulated LSAT questions and are under development. Question revision will be complete by Sept. 21st.
The first step of diagramming conditionals is to first determine whether a phrase is a sufficient or a necessary condition. The key here is to look for the conditional indicators.
The following is a list of LSAT conditional indicators to help you determine whether a phrase is a sufficient or a necessary condition.
- If, If only
- All, Any
- Each, Every
- When, Whenever, Whoever, Whatever, Whenever
- People who
- In order
- Only, Only if, Only when
- Relies on, Depends on
Steps for diagramming conditionals
For you to more easily analyze the questions that will be given, it is important that you use a system of steps that will help you diagram conditions.
The following is a proven and effective way to correctly diagram conditionals:
- Locate the conditional indicator of the statement. Locate the sufficient indicator if it is a sufficient condition or locate the necessary indicator if it is a necessary condition.
- Locate the conditional phrase after the conditional indicator. If it is a sufficient indicator, the conditional phrase that follows it is the sufficient condition. But if it is a necessary indicator, the conditional phrase that follows it is the necessary condition.
- Abbreviate the conditional phrase so it becomes clearer and easier to diagram. Our suggestion is to abbreviate the phrase using at least one letter and a maximum of three letters.
- Locate the conditional result in the conditional statement. This is the phrase that is left when you abbreviate the conditional phrase after the conditional indicator. If the statement has a sufficient indicator, the conditional result is the necessary phrase. If the statement has a necessary indicator, the conditional result is the sufficient phrase.
- Abbreviate the conditional result so it becomes clearer and easier to diagram. Similarly, our suggestion is to abbreviate the conditional result using at least one letter and a maximum of three letters.
- Finally, combine the two conditional phrases in the form: sufficient phrase → necessary phrase.
The negation symbol/tilde symbol (~)
There are certain conditional indicators that require special attention and the use of the negation symbol (~).
- We use (~) when the sufficient indicator is “no/none” or “never”. If one of these sufficient indicators is present, it means that we should first negate the necessary condition using (~) before combining both the sufficient and necessary phrases in the form: sufficient phrase → necessary phrase.
- The necessary indicators “unless”, “until”, and “except” also require the use of the negation symbol (~). If one of these necessary indicators is present, it means that we should negate the sufficient condition using (~) before combining both the sufficient and necessary phrases in the form: sufficient phrase → necessary phrase.
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.
It is very easy to diagram the contrapositive of a conditional statement if you already know the diagram of the original statement. You just have to negate the conditional phrases using (~) and switch their positions.
- 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. Furthermore, since the contrapositive of a conditional statement is just another conditional statement, all the rules apply the same.
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.
From the previous chapter, we learned that we can validly infer the contrapositive from the conditional statement itself. Using our knowledge of contrapositives, we can now combine conditional statements to form a compound statement. The combination of a conditional statement and its contrapositive form the statement “If not A → B”.
- Premise 1: ~A → B
- Premise 2 (the contrapositive of premise 1): ~B → A
- Conclusion: (~A → B) and (~B → A), or If not A → B
“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).
The idea behind a biconditional statements is that the sufficient condition of a biconditional statement is also a necessary condition, and its necessary condition is also a sufficient condition.
From the biconditional A ↔ B, we can infer the following statements:
- A → B
- B → A
- (A → B) and (B → A)
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)
The transitive property comes in the following form:
If A → B and B → C, then A → C
“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). However, there are some slight changes since one of the conditions is a compound phrase.
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 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
An invalid inference occurs when you incorrectly infer a statement form another conditional statement. This error is very common, especially to those who are not well-trained on formal logic.
Invalid inferences are only concerned with the form of conditional statements and not their content. The premises of a conditional statement may be true in real life, but the statement itself may still be invalid.
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.
Remember that you can validly infer only the contrapositive of a conditional statement and not its converse.
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. You cannot infer the inverse of a conditional statement from itself because they are logically different statements.
Remember that you can validly infer only the contrapositive of a conditional statement and not its inverse or converse.
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:
Previously, we only dealt with statements containing the qualifiers “no/none”, “all” and their variations. However, there are also other qualifiers you should know about. Understanding these qualifiers and how to represent them is important for passing the LSAT logic section. The other qualifiers are “some” and “some are not”.
- All means the 100% of something.
- “Some” means greater than 0% but less than 100%. In other words, it ranges from 1% to 99%.
- “Some are not” means less than 100%, but unlike “some”, it also includes 0%.
Take note: The opposite/negation of “all” is NOT “none”. The opposite/negation of “all” is “some are not”. Always remember not to confuse these because a lot of people make this common mistake, which can easily be avoided by always paying close attention.
Furthermore, these qualifiers follow the same principles from the previous ones. This means that they are not reversible – the relations of these statements and their contrapositive, converse, and inverse are similar to that in Figure 1.
The main difference between informal and formal logic is that the former deals with the content of an argument and usually with ‘fuzzy logic’, while the latter deals with the logical structure of an argument. Most LSAT logic questions will 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” type of questions is to diagram them and identify their parts and relations, similar to what we did in the previous chapters. Form valid inferences like finding the contrapositive of a statement and applying the transitive property so you can maximize the number of information available to you. Moreover, always be aware of invalid inferences so you can avoid them.
This type of questions in the LSAT will have a total of five choices. Four of the choices in a “must be true” questions can be correct, which means that they can be wrong or are only likely but not guaranteed to be true; but only one of these questions must be correct, which means that it is always true given the information provided by the question, and this is the correct answer.
Must be false
Likewise, “must be false” type of questions can be more easily solved by diagramming them and identifying their parts and relations among these types. Forming valid inferences would also be very helpful.
Similar to “must be true” type of questions, this type of questions will have a total of five choices in the LSAT. 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.
Both ‘must be true’ and ‘must be false’ questions can be solved more easily by elimination.
“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. This type of questions is similar to “must be true” type of questions but they differ in the degree of certainty that an answer can be true. If the question asks something that must be or absolutely be true, then it is a “must be true” question. On the other hand, if the question asks something most supported by the given information but does not have to be necessarily absolutely true, then it is a “most strongly supported” question.
Steps on solving “most strongly supported” questions
- Similar to any problem concerning inference, the first thing you have to do is look for the relation among the statements.
- Analyze the logical structure of the set of premises and search for language cues to determine what kind of reasoning the question has.
- Anticipate how the answer would sound like given these set of premises and knowing their relations.
- Finally, review the answers and see which one best fits the given premises.
Tips for solving this “most strongly supported” type of questions
- As with most other logic questions, the first possible step to further clarify and analyze a set of premises is to diagram them.
- You have to read and understand the question carefully and go through all the answer choices attentively to get a better idea of which of the answer choices is most strongly supported.
- 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. Thus, some answers may not have very strong support from the given premises. The important thing is that it is the most strongly supported among the choices.
- Always be wary of invalid inferences and trap choices.
Congratulations! You’ve finished our Formal Logic chapter. You should know the basics of conditional logic. In the real world, however, things are rarely so simple as cut-and-dry conditional statements. Arguments are packed with assumptions. You’re moving on to the woolly world of informal logic!
Next LSAT: September 21st
Next LSAT: September 21st