The easiest inference in formal logic is called the Transitive Property:
If A → B, and B → C; then A → C.
(you can eliminate B)
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).
What happens when you try to contrapose conditionals joined by conjunctions (and, or)?
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.
- 0:12 – Let us take a look at this statement: If you flip the switch and the computer’s plugged in then the computer will turn on.
- 0:27 – We can 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 – We may 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 – In general, when you have an AND in the necessary condition, the AND turns into an OR in the contrapositive.
- 2:40 – Let us take a look at 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. We can diagram this as: ~Amber AND ~Sergio → Lulu home.
- 3:02 – Note: The “neither nor” should be interpreted similarly to “and” and not “or”.
- 3:54 – The contrapositive of this statement can be diagrammed as: ~Lulu home → Amber OR Sergio.
- 4:48 – Note: Remember that 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 – Let us take a look at another example: If Khaled is on the fencing team then he tried out for the fencing team and was accepted. We can diagram this as: team → tried out AND accepted.
- 5:15 – We can think of this as 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.
When two conditionals are joined by an or, the negation is now an and.
- 0:09 – Let us take a look at this conditional statement: If Jane plays the guitar or the piano then Jane plays an instrument.
- 0:20 – We can infer from this that either element in the sufficient condition works as a guarantee. We can 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 – We can then 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 – Let us take a look at 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 – We can think of it as two conditionals joined by OR. We can diagram the conditional as follows: building → engineer OR journalist.
- 2:33 – We can then 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 – We diagram the contrapositive as follows: ~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: October 28th
“If my mind can conceive it, and my heart can believe it – then I can achieve it.”
– Muhammad Ali
“Only when your desires are distilled, will you love more and be happy.”
Magician of Contraposition
- 0:19 – We can find conditional chains in formal logic questions in logical reasoning as well as logic games that’s why it is important to familiarize yourself with them.
- 0:24 – Let us 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 – We can diagram the first statement as: C → B.
- 0:44 – We can also 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 – This can also be done with the contrapositives. The contrapositive of the first statement is: ~B → ~C. The contrapositive of the second statement is: T → ~B.
- 1:25 – These statements again 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 – Let us take a look at a more difficult example in a logic game set up. Consider the situation where part invitations are going out. The potential guests are Mike, Andrew, Jon, Steve, Willy, and Todd.
- 1:54 – The stipulations then 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 Todd. If we don’t invite Mike, then we’ll have to invite Willy.
- 2:08 – We diagram these statements as follows:
- 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 – We can then 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 – We can form the following conditional from these: A → J → ~T → ~W → M.
- 3:20 – This can also be done with the contrapositive which is as follows: ~M → W → T → ~J → ~A.
- 3:30 – We can then use this to answer certain questions about this. For example: If Todd is not invited then we can infer that Mike is not invited. We can use the original conditional to answer this.
- 3:49 – Similarly, if you are asked to choose among choices which one is true, you just have to refer to the conditional or its contrapositive. For example, choice (B) W → ~A in the given question can be found in the contrapositive.
- 4:29 – Another place where we can find these conditional chains is in logical reasoning.
- 4:53 – We can diagram the statements in the example provided in the video as follows: 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 – We can solve this question by considering the contrapositive of the statement we wrote, which is as follows: ~admissions chance → ~transfer AND ~repeat → ~Eddie fails → ~difficulty increases. Therefore, the answer to the question is when the difficulty did not increase (~difficulty increase).
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. This isn’t as hard as it looks once you get comfortable with it.
Conditionals w/Conjunctions Drills
If A’s delivery is earlier than B’s, then C’s delivery is earlier than D’s.
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.
Whenever I am on the road, I get stomach aches and I can’t get Swedish meatballs.
Sherlock Holmes on Inferences
Every problem is absurdly simple when it is explained to you.
- 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.
- 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.
- Note: There are still some hidden inferences that Sherlock did not explicitly state. These are as follows:
- Meeting Thurston means Watson was asked by Thurstone if he agrees to invest. Diagram: Thurston → Watson asked to invest.
- If Watson was asked, he would either agree or not. Diagram: Watson asked to invest → agree OR Watson asked to invest → ~agree.
- If Watson agrees, then he would invest. But if he does not agree, then he would not invest. Diagram: agree → invest OR ~ agree → ~invest.
- If Watson agreed, then he would take the checkbook from Sherlock’s drawer. Diagram: agree → ~checkbook locked in drawer (the checkbook would no longer be locked in the draw).
- We take the contrapositive of the previous conditional, which means that if the checkbook is still locked in the drawer, then Watson did not agree. Diagram: checkbook locked in drawer → ~ agree.
- We know that the checkbook is still locked in the drawer as Sherlock mentioned. Therefore, Watson did not invest. Diagram: checkbook locked in drawer → Watson disagree → Watson did not 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).