The Logic of If vs. Only If — Khan Academy

Source: Khan Academy / LSAT prep series Raw file: raw/The Logic of If vs. Only if - Khan Academy.md

Summary

"If" and "only if" sound nearly identical but point in opposite directions. This is one of the most exploited distinctions on the LSAT. The source works through the hat/sunny example and then lists four surface variations of "only if" that all reduce to the same logical structure.

The Core Contrast

"I wear a hat if it's sunny."

Another reading: "If it's sunny, then I wear a hat."

Diagram: sunny → hat

Sunny is sufficient to guarantee hat-wearing. On 100% of sunny days, I wear a hat.

Contrapositive: ¬hat → ¬sunny

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"I wear a hat only if it's sunny."

This does NOT mean sunniness guarantees hat-wearing. It means hat-wearing guarantees sunniness — I would only ever wear a hat on a sunny day.

Diagram: hat → sunny

Hat-wearing is sufficient to guarantee that it is sunny.

Contrapositive: ¬sunny → ¬hat

What does NOT follow: sunny → hat (it might be sunny and I don't wear a hat) or ¬hat → ¬sunny (it might be sunny even if I'm hatless).

Summary Table

"Only if" points to the necessary condition — but the necessary condition is always on the right side of the arrow (the result), never the trigger.

"Only" Variations — All the Same

These four phrasings are logically identical:

• I only wear a hat if it's sunny.
• I wear a hat only when it's sunny.
• The only time I wear a hat is if it's sunny.
• Only sunny days will get me to wear a hat.

All diagram as: hat → sunny

Top tip: Rephrase any "only" statement as either "X only if Y" or "If X, then Y" before diagramming. Don't let word order fool you — figure out what is required vs. what guarantees another event.

Concepts

• sufficient-and-necessary-conditions — the "if" vs "only if" distinction is the source's single teachable unit; it extends the concept page's signal-word table with an explicit contrast between the two phrasings