Conditional Reasoning: Converse and Contrapositive

Methods in this video

Tap a time to jump to where the video teaches that method.

1. 1

   Draw the arrow: follow it, never backwards

   ▶ 0:49

   When to use it: The box gives an "if... then..." rule (if, when, whenever, always), one person reasons from the second part back to the first ("he isn't at school, so he must have a headache"), and the question asks which sentence shows the mistake.

   • Draw "if A then B" as an arrow A → B. It is a one-way road: follow it, never go backwards.
   • Reasoning from B back to A is going backwards. To show the mistake, say "B could happen for another reason."
   • Words like whenever and always in a question give you arrows too.
2. 2

   Four statements, two that work

   ▶ 2:30

   When to use it: The box gives an "if... then..." rule, and someone reasons from "not the second part" to "not the first part", or from "not the first" to "not the second". You must decide whether it works; most often it is a two-person question (whose reasoning is correct?).

   • One arrow A → B gives four statements: A → B, B → A, no B → no A, and no A → no B.
   • Only two of them work: following the arrow (A → B), and no B → no A.
   • Going back is not always wrong: going back when both ends say "no" works.
3. 3

   "Only": draw the arrow the other way

   ▶ 4:31

   When to use it: The rule says "only" (only those who..., the only times...) or "you need at least... to have a chance", and someone concludes "he did it, so he definitely can" or "he didn't, so he can't".

   • Draw "only A can B" as B → A: what comes after "only" is where the arrow points.
   • Having A does not guarantee B; without A, there is definitely no B.
   • Reading "only" as "if" makes you go backwards.
4. 4

   Join the rules into a chain and follow the arrows

   ▶ 5:59

   When to use it: Two or three rules join end to end (everyone who likes A likes B, everyone who likes B likes C but no one likes D), and the question asks which statement must be true.

   • Join rules end to end into one chain: A → B → C. A chain can branch.
   • Start where the option starts and follow the arrows. If you can get there, it must be true.
   • Going backwards, starting from "no A", or starting from where an arrow points are all wrong turns.