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What happens when you try to contrapose conditionals with embedded conjunctions or disjunctions?
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 
When two conditionals are joined by “or,” the negation becomes “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: August 14/15
“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
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.
(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.
(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
“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
If A’s delivery is earlier than B’s, then C’s delivery is earlier than D’s.
(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.
~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.
IF on the road → stomach aches and can’t get Swedish meatballs
IF NOT stomach aches or get Swedish meatballs → NOT on the road
This is an adaptive drill: The questions will get harder or easier depending on your performance. You can't go backwards or change prior answers.
Complete: 0 / 3 correct
Every problem is absurdly simple when it is explained to you.
Sherlock Holmes
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.
(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.
(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.
(C) If the country’s agriculture is not doing good, then the country’s GDP is not increasing.
Statement 
Symbols 
Valid?/ 

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 
