LSAT Course > Logical Reasoning > Formal Logic > Conditional Conjunctions

Conditional Conjunctions

What happens when you try to contrapose conditionals joined by conjunctions (and, or)?

"And" Conditionals

Conditional statements sometimes have multiple entities for the sufficient or necessary joined by an “and“. When negating you have to turn the “and” into an “or“.

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 neither 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 to be 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 an AND, we need to turn it into an OR.

"Or" Conditionals

When two conditionals are joined by an “or“, the negation is now an “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 turn it into an AND.

Next LSAT: January 13th

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
thinks 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

Conditional Conjunctions Examples

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.
(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.
(B) If Charm does not play basketball and does not play volleyball, then she is not athletic.

Answer

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

Statement

Symbols

Valid/Invalid

Description

1. If Charm plays basketball or volleyball then she is athletic.

basketball OR volleyball → athletic

Given

Given

2. If Charm is not athletic then she does not play basketball and she does not play volleyball.

~athletic → ~basketball AND ~volleyball

Valid

Contrapositive

3. If Charm does not play basketball and does not play volleyball, then she is not athletic.

~basketball AND ~volleyball → ~athletic

Invalid

Inverse

4. If Charm 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 an 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.

Contrapositive

(A – B) → (C – D)

~(C – D) → ~(A – 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

IF NOT G → E
IF NOT E → G

Either E or G or both at all times

IF NOT G and NOT E → Leave
IF NOT Leave →  G or E

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

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