Conditional Reasoning and Logical Equivalence — Khan Academy
Conditional Reasoning and Logical Equivalence — Khan Academy
Source: Khan Academy / LSAT prep series
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Summary
Identifies the four variations of a conditional statement and shows which are logically equivalent and which are not. Only the contrapositive is always logically equivalent to the original. The converse and inverse are not — confusing them with valid inferences is one of the most common logical errors.
The Four Variations
Given: A → B (e.g., "Whenever I do yoga, I feel calm")
| Name | Form | Logically equivalent? |
|---|---|---|
| Original | A → B | — |
| Contrapositive | ¬B → ¬A | Yes |
| Converse | B → A | No |
| Inverse | ¬A → ¬B | No |
Why the contrapositive is valid: If A guarantees B, then not-B is proof that A couldn't have happened. Reverse the arrow and negate both sides.
Why the converse is invalid: "If I'm calm, then I'm doing yoga." Calm has many other causes. Knowing the result doesn't prove the trigger.
Why the inverse is invalid: "If I'm not doing yoga, I'm not calm." Not doing yoga still allows other triggers of calm (sleep, vacation, a puppy). Absence of one sufficient condition doesn't eliminate the result.
Forming a Contrapositive with Multiple Terms
When the original has compound conditions (AND or OR), the contrapositive requires an extra step.
Example: "If I'm skateboarding, I will wear a helmet and protective gloves."
Diagram: Skateboarding → helmet and gloves
Step 1 — Flip: Move everything on the left to the right and vice versa.
Helmet and gloves → skateboarding (not yet equivalent)
Step 2 — Negate every term:
¬helmet and ¬gloves → ¬skateboarding (not yet equivalent)
This reads "if I'm wearing neither," which is too narrow — if I have gloves but no helmet, I'm still not skateboarding.
Step 3 — Flip AND ↔ OR:
¬helmet or ¬gloves → ¬skateboarding (now equivalent)
If either piece of necessary equipment is missing, I cannot be skateboarding.
The rule: In any contrapositive, every AND becomes OR and every OR becomes AND.
Concepts
- sufficient-and-necessary-conditions — names and defines the four variations; adds the three-step multi-term contrapositive procedure and the AND↔OR flip rule