LSAT Course > Logical Reasoning > Formal Logic > Conditional Chains

Combine Conditionals

The easiest inference in formal logic is called the Transitive Property:

If A → B, and B → C; then A → C.

Valid Inference (Transitive Property):
A → C

If A, then B: If I press the off button (A), then the generator will turn off (B).

If B, then C: If the generator is off (B), then the website will shut down (C).

If A, then C: If I press the off button (A), then the website will shut down (C).

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Next LSAT: January 13th

Magician of Contraposition

Now things are going to get more challenging. We’re going to:
1) Make conditionals
2) Contrapose them
3) Look for conditionals with common values that we can link together and use the transitive property to make more inferences.

So, a simple series of statements can produce an enormous number of inferences.

This video consolidates everything we’ve covered. Watch how he contraposes conditionals joined with “and.” He pulls inferences like rabbits out of a hat.

This is a challenging video so you might want to watch it twice to understand. But this isn’t as hard as it looks once you get comfortable with it.

Creating Conditional Chains Video Summary

  • 0:24 – Consider these two statements: If Charlie makes the soccer team, then Bill will make the soccer team. If Bill makes the soccer team, then Tom cannot make the team.
  • 0:37 – Diagram the first statement as: C → B.
  • 0:44 – Diagram the second statement as: B → ~T.
  • 0:52 –These two statements share the common element of Bill (B). This means that we can combine them to become: C → B → ~T. This implies that therefore, if Charlie makes the team, then Tom cannot. Its diagram is: C → ~T.
  • 1:12 – The contrapositive of the first statement is: ~B → ~C. The contrapositive of the second statement is: T → ~B.
  • 1:25 – These statements share the common element of Bill (B) so we can also combine them to become: T → ~B → ~C. Therefore, if Tom makes the team then Charlie cannot. Its diagram is: T → ~C.
  • 1:44 – Consider a more difficult example. Party invitations are going out and the potential guests are Mike, Andrew, Jon, Steve, Willy, and Todd.
  • 1:54 – The stipulations are as follows: If Todd is invited, Jon won’t be invited. Inviting Willy means we have to invite Todd and Steve. We’ll invite Andrew only if we also invite Jon. If we don’t invite Mike, then we’ll have to invite Willy.
  • 2:08 – Diagram these statements as:
    • T → ~J; contrapositive: J → ~T
    • W → T and S; contrapositive: ~T or ~S → ~W
    • A → J; contrapositive: ~J → ~A
    • ~M → W; contrapositive: ~W → M
  • 2:52 – Combine these conditionals by identifying their common elements (2 T’s, 2 ~J’s, 2 J’s, 2 ~T’s, 2 W’s, and 2 ~W’s).
  • 2:57 – Form the following conditional chain: A → J → ~T → ~W → M.
  • 3:20 – Do the same with the contrapositives: ~M → W → T → ~J → ~A.
  • 3:30 – For example: If Todd is not invited then we can infer that Mike is not invited. Use the original conditional to answer this.
  • 3:49 – To choose which one is true, just refer to the conditional or its contrapositive. For example, choice (B) W → ~A in the given question can be found in the contrapositive.
  • 4:53 – Diagram the statements in the example: difficulty increases → Eddie fails → transfer OR repeat → admissions chance decrease (since transfer → admissions chance decrease OR repeat → admissions chance decrease).
  • 5:25 – Consider the question: What must be true if Eddie’s chances of college admissions did not decrease?
  • 5:38 – Solve this question by using the contrapositive of the statements: ~admissions chance → ~transfer AND ~repeat → ~Eddie fails → ~difficulty increases. Therefore, the answer to the question is when the difficulty did not increase (~difficulty increase).

Sherlock Holmes Explains a Conditional Chain

Every problem is absurdly simple when it is explained to you.
-Sherlock Holmes

160 Seconds of Inferences

  • 0:16 – Sherlock Holmes concludes that Watson does not want to invest in South African securities.
  • 0:45 – What Sherlock did was simply remove these central inferences and present only the starting point and the conclusion (meretricious means flashy, show off in this context).
  • 1:05 – Through the inspection of the groove between Watson’s left forefinger and thumb, Sherlock concludes that Watson has decided not to invest his small capital in the gold fields.
  • 1:21 – Sherlock explains the connections as follows:
    • 1:22 – Watson had chalk between his forefinger and thumb when he returned from the club last night. Diagram as: chalk between thumb → play billiards.
    • 1:30 – Watson never plays billiards except with Thurston. Diagram as: play billiards → Thurston.
    • 1:34 – Thurston had an option on some South African security which expired in a month and which he desired to share with Watson.
    • 1:42 – Watson’s checkbook is still locked in Sherlock’s drawer and he has not asked for the key. Diagram: checkbook locked in drawer → ~invest.
  • The complete conditional chains then are as follows:
    • chalk between thumb → plays billiards → Thurston → Watson asked to invest → (agree → invest) OR (~agree → ~invest)
    • checkbook locked in drawer → ~agree → ~invest
    • Therefore, Watson turned down the investment.
  • 2:09 – Watson then makes his own inference. If Sherlock has no cases, then he is in black moods (not cheerful). Diagram: no cases → ~cheerful. We can infer the contrapositive as: cheerful → ~no cases (has a case).
  • 2:25 – However, since Sherlock is cheerful, then he must have a case. Diagram: cheerful → ~no case (has a case).

Conditional Chain Example

If Jean went to the concert, then Ricardo also went to the concert. If Ricardo went to the concert, then Adrian also went to the concert. Jean went to the concert.

If the above statements are true, which of the following must also be true?
(A) If Jean did not go to the concert, then Adrian also did not go to the concert.
(B) Adrian also went to the concert.

Answer

(B) Adrian also went to the concert.

Statement

Symbols

Valid/Invalid

Description

1. If Jean went to the concert, then Ricardo also went to the concert.

J → R

Given

Premise 1

2. If Ricardo did not go to the concert, then Jean did not go to the concert.

~R → ~J

Valid

Contrapositive of premise 1

3. If Ricardo went to the concert, then Adrian also went to the concert.

R → A

Given

Premise 2

4. If Adrian did not go to the, then Ricardo did not go to the concert.

~A → ~R

Valid

Contrapositive of premise 2

5. If Jean went to the concert, then Adrian also went to the concert.

J → A

Valid

Inference from premise 1 and 2

6. If Jean did not go to the concert, then Adrian also did not go to the concert.

~J → ~A

Invalid

Inverse of inference from premise 1 and 2

Due to a conflict in schedule, I can watch movie 1 only if I do not watch movie 2. If I don’t watch movie 3, then I can watch movie 2.

If the above statements are true, which of the following must also be true?
(A) If I watch movie 3 then I can watch movie 1.
(B) If I can watch movie 1 then I can also watch movie 3.
(C) If I watch movie 3 then I can’t watch movie 2.

Answer

(B) If I can watch movie 1 then I can also watch movie 3.

Statement

Symbols

Valid/Invalid

Description

1. I can watch movie 1 only if I do not watch movie 2.

M1 → ~M2

Given

Premise 1

2. If I don’t watch movie 3, then I can watch movie 2.

~M3 → M2

Given

Premise 2

3. If I can watch movie 1 then I can also watch movie 3.

M1 → M3

Valid

Inference from Premise 1 and 2

4. If I watch movie 3 then I can watch movie 1.

M3 → M1

Invalid

Converse of Inference

5. If I watch movie 3 then I can’t watch movie 2.

M3 → ~M2

Invalid

Inverse of Premise 2

Increase in the country’s GPD means the country’s economy is improving. If the country’s agriculture is not doing good, then the country’s economy is not improving.

If the above statements are true, which of the following must also be true?
(A) If the country’s agriculture is doing good, then the country’s economy is improving.
(B) If the country’s agriculture is doing good, then the country’s GDP is increasing.
(C) If the country’s agriculture is not doing good, then the country’s GDP is not increasing.

Answer

(C) If the country’s agriculture is not doing good, then the country’s GDP is not increasing.

Statement

Symbols

Valid/Invalid

Description

1. Increase in the country’s GPD means the country’s economy is improving.

GDP → economy

Given

Premise 1

2. If the country’s agriculture is not doing good, then the country’s economy is not improving.

~agriculture → ~economy

Given

Premise 2

3. If the country’s agriculture is not doing good, then the country’s GDP is not increasing.

~agriculture → ~GDP

Valid

Inference from Premise 1 and 2

4. If the country’s agriculture is doing good, then the country’s economy is improving.

agriculture → economy

Invalid

Inverse of Premise 1

5. If the country’s agriculture is doing good, then the country’s GDP is increasing.

agriculture → GDP

Invalid

Inverse of Inference

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