### Conjoin Conditionals

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)?

#### "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.

#### "Or" Conditionals

When two conditionals are joined by an or, the negation is now an and.

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

### Contrapositive

The red text shows the 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

### Contrapositive

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See contrapositive in red. And becomes Or.
LM and H → DD
~DD → ~(LM or H)

#### Magician of Contraposition

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.

#### Conditionals w/Conjunctions Drills

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

### Contrapositive

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

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

### Contrapositive

#### Sherlock Holmes on Inferences

Every problem is absurdly simple when it is explained to you.

### Watch Sherlock Holmes and Watson apply conditional reasoning (optional).

#### 200 seconds of blazing inferences

SHERLOCK
0:16
Conclusion: Watson does not want to invest in South African securities.
1:06 Chalk between thumb to ease cue => play billiards.
1:30 Billiards => Thurston. Thurston => ask to invest in expiring option in South Africa.
1:44 Did not ask for checkbook => not invest in expiring option.
Therefore, he turned down the investment

WATSON:
2:09 No cases => Sherlock homes has “black moods”
Contrapositive
2:29 Sherlock Holmes is cheerful => has a case