LSAT Course > Logical Reasoning > Formal Logic > Conditional Conjunctions

#### Disjunctions and conjunctions within conditionals

What happens when you try to contrapose conditionals with embedded conjunctions or disjunctions?

### Conjunctions within conditionals

Note: By convention, if we omit brackets, as in

A and/or B → C

we read it as

(A and/or B) → C

and not as “A and/or (B → C)”. Similarly

A and/or B ↔ C

is read as

(A and/or B) ↔ C

Also by convention, negation applies only to the sentence that occurs immediately after the negation sign. Thus

~A or/and B
~A → B
~A ↔ B

are read as

(~A) or/and B
(~A) → B
(~A) ↔ B

and not as “~(A or/and B)”, “~(A → B)”, “~(A ↔ B)”.

We know that a conjunction is true when both clauses (the “conjuncts”) are true. When is a conjunction false?
A conjunction is false if and only if at least one conjunct is false. We will express this using symbols:

~ (A and B)
is equivalent to
~A or ~B

A disjunction is false if and only if both disjuncts are false:

~ (A or B)
is equivalent to
~A and ~B

When we know that, and we know the rule of double negation, we can derive other versions of the above rules, for example:

A and B
is equivalent to
~ ( ~A or ~B )

~ (~A and B)
is equivalent to
A or ~B

A and ~B
is equivalent to
~ ( ~A or B )

A or B
is equivalent to
~ ( ~A and ~B )

~ (A or ~B)
is equivalent to
~A and B

A or ~B
is equivalent to
~ ( ~A and B )

Back from symbols into English:

 A and B means both A and B A or B means at least one of A and B ~ (A and B), ~A or ~B mean not both A and B ~ (A or B), ~A and ~B mean neither A nor B

### Multiple “AND” Conditions Video Summary

• 0:12 – Consider this statement: If you flip the switch and the computer’s plugged in then the computer will turn on.
• 0:27 – Diagram this as follows: switch AND plugged in → turn on.
• 0:34 – This means that the elements in the sufficient condition work together as a guarantee for the necessary condition. Having just one of the elements doesn’t mean that the necessary condition has to follow.
• 1:23 – If the necessary condition isn’t true, then either of the two elements in the sufficient condition may be false, not necessarily both of them.
• 1:29 – Diagram this as follows: switch AND plugged in → turn on; its contrapositive is: ~turn on → ~switch OR ~plugged in.
• 1:57 – If the necessary condition isn’t true then the computer won’t turn on.
• 2:15 – When you have an AND in the necessary condition, the AND turns into an OR in the contrapositive.
• 2:40 – Another example: If neither Amber nor Sergio goes to the concert then Lulu will stay home.
• 2:50 – The “neither nor” should be interpreted as AND where the two elements are both negative. Diagram this as: ~Amber AND ~Sergio → Lulu home.
• 3:02 – Note: The “neither nor” should be interpreted similarly as “and” and not “or”.
• 3:54 – The contrapositive of this statement can be diagrammed as: ~Lulu home → Amber OR Sergio.
• 4:48 – Note: When negating for the contrapositive, change the AND to OR.
• 4:54 – The contrapositive of the statement becomes: Lulu not staying at home means that either Amber went to the concert or Sergio went to the concert.
• 5:05 – Consider another example: If Khaled is on the fencing team then he tried out for the fencing team and was accepted. Diagram this as: team → tried out AND accepted.
• 5:15 – These are like two conditionals. If Khaled is on the team then he tried out; If he’s on the team then he was accepted. The diagrams are as follows: team → tried out; team → accepted.
• 5:53 – If the necessary condition isn’t true, this means that either one of the two elements is false, but not necessarily both of them.
• 6:03 – Since both elements are requirements, you only need to have one requirement missing for the sufficient condition to be false.
• 6:11 – Its contrapositive can be diagrammed as: ~tried out OR ~accepted → ~team.
• 6:43 – Recap: A conditional statement can have more than one element in the sufficient or necessary condition joined by AND. When negating AND, we need to change it to OR.

#### Conditionals and disjunctions

When two conditionals are joined by “or,” the negation becomes “and.”

### Multiple “OR” Conditions Video Summary

• 0:09 – Consider this conditional statement: If Jane plays the guitar or the piano then Jane plays an instrument.
• 0:20 – Infer from this that either element in the sufficient condition works as a guarantee. Diagram this as: guitar OR piano → instrument.
• 0:37 – Having just one of the elements means that the necessary condition has to follow.
• 0:42 – Infer the following conditionals: guitar → instrument OR piano → instrument.
• 1:01 – Having an OR in the sufficient condition is the same as having two conditionals joined by OR.
• 1:10 – If the necessary condition isn’t met, then neither one of the elements in the sufficient condition can be true.
• 1:22 – Its contrapositive can be diagrammed as follows: ~instrument → ~guitar AND ~piano.
• 1:58 – Another example: If Ang works in that building then he is an engineer or he is a journalist.
• 2:08 – If the sufficient condition is true, either element in the necessary condition could be true. The sufficient condition is a guarantee for either one of the two elements.
• 2:19 – Think of it as two conditionals joined by OR. Diagram the conditional as follows: building → engineer OR journalist.
• 2:33 – Diagram this as two different conditionals: building → engineer OR building → journalist.
• 2:55 – If neither of the elements in the necessary condition is true, then the sufficient condition can’t be true. At least one of the requirements must be met.
• 3:12 – Diagram the contrapositive as: ~engineer AND ~journalist → ~building.
• 3:59 – Recap: A conditional statement can have more than one element in the sufficient condition or more than one element in the necessary condition joined by OR. When negating an OR, we need to change it to an AND.

A → B and C
is equivalent to the conjunction
(A → B) and (A → C)

A → B or C
is equivalent to the disjunction
(A → B) or (A → C)

but pay attention to this one:

A or B → C
is equivalent to the conjunction
(A → C) and (B → C)

A and B → C cannot be expressed in terms of simple conditionals in a similar way. But this relation is useful to know:

A and B → C
is equivalent to
A → (B → C)

Next LSAT: Jun 10/Jun 11

### Video Summary

“Anyone who thinks science is trying to make human life easier or more pleasant is utterly mistaken.”

Diagram this as two separate conditional statements:
think science tries to make life easier → utterly mistaken
think science tries to make life pleasant → utterly mistaken

Infer their respective contrapositives:
~utterly mistaken → ~think science tries to make life easier
~utterly mistaken → ~think science tries to make life pleasant

#### Examples of conditionals with embedded conjunctions

If you pass your exams and complete the requirements, then you pass the subject. You did not pass the subject.

If the above statements are true, which one of the following must be true?
(A) You did not pass the exams.
or
(B) You did not pass the exams and did not complete the requirements.

### Answer

(A) You did not pass the exams.

 Statement Symbols Valid/Invalid Description 1. If you pass your exams and complete the requirements, then you pass the subject. pass exams AND complete requirements → pass subject Given Given 2. If you did not pass the subject, then you either did not pass the exams or did not complete the requirements. ~pass subject → ~pass exams OR ~complete requirements Valid Contrapositive 3. If you do not pass the exams or do not complete the requirements, then you do not pass the subject. ~pass exams OR ~complete requirements → ~pass subject Invalid Inverse 4. If you passed the subject, then you passed the exams and completed the requirements. pass subject → pass exams AND complete requirements Invalid Converse

If Charm plays basketball or volleyball then she is athletic.

If the above statements are true, which one of the following must be true?
(A) If Charm is not athletic then she does not play basketball and she does not play volleyball.
or
(B) If Charm does not play basketball and does not play volleyball, then she is not athletic.

### Answer

(A) If Sarah is not athletic then she does not play basketball and she does not play volleyball.

 Statement Symbols Valid/Invalid Description 1. If Sarah plays basketball or volleyball then she is athletic. basketball OR volleyball → athletic Given Given 2. If Sarah is not athletic then she does not play basketball and she does not play volleyball. ~athletic → ~basketball AND ~volleyball Valid Contrapositive 3. If Sarah does not play basketball and does not play volleyball, then she is not athletic. ~basketball AND ~volleyball → ~athletic Invalid Inverse 4. If Sarah is athletic then she plays basketball or she plays volleyball. athletic → basketball OR volleyball Invalid Converse

“If my mind can conceive it, and my heart can believe it – then I can achieve it.”
– Muhammad Ali

### Contrapositive

“If my Mind Can Conceive it, and my Heart can Believe It – then I can Achieve It.”

MCI & HBI → AI

~AI → ~MCI or ~HBI (change the and to or)

“Only when your desires are distilled, will you love more and be happy.”
-Hafiz

### Contrapositive

If A’s delivery is earlier than B’s, then C’s delivery is earlier than D’s.

### Contrapositive

(A before B) → (C before D)

~(C before D) → ~(A before B)

If Boris can’t find Groucho, he’ll get Erin instead. If Boris can’t get Groucho and can’t get Erin, then he will move on to the next store.

### Contrapositive

~Groucho → Erin
~Groucho and ~Erin → Next

~Erin → Groucho
~Next → Groucho or Erin

If Boris won’t move on to the next store, then this means that he found Groucho or Erin.

Whenever I am on the road, I get stomach aches and I can’t get Swedish meatballs.

### Contrapositive

IF on the road → stomach aches and can’t get Swedish meatballs

IF NOT stomach aches or get Swedish meatballs → NOT on the road

# Chains of conditionals

### 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 Logic

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).

#### Examples with chains of conditionals

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 concert, 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 GDP 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?/ Description 1. Increase in the country’s GDP 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