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.”
– Muhammad Ali


The red text shows the contrapositive.

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


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

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



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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.

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.

200 seconds of blazing inferences

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

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


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