A liar who always lies says "All my hats are green."

Original Article ↗ Hacker News Discussion ↗

Formal-logic reading and the “correct” answer

  • Many participants translate “All my hats are green” into a universal quantification over the set of the liar’s hats.
  • Under classical logic, a universal statement over an empty set is vacuously true.
  • Since the liar “only says false statements”, the utterance must be false, so the set of hats cannot be empty.
  • Negating “All my hats are green” yields “There exists at least one of my hats that is not green.”
  • From this, people conclude:
    • The liar has at least one hat.
    • At least one of those hats is not green.
    • Among the multiple-choice options, only “The liar has at least one hat” is necessarily true.

Colloquial / linguistic objections

  • Others argue that in everyday English, “all my X” strongly implies “I have at least one X”, so with zero hats the statement is already a lie.
  • On that reading, the lie could be:
    • about existence (having no hats), or
    • about color (having non‑green hats), or both.
  • Therefore none of the answer choices are forced; multiple scenarios are consistent.
  • This leads into discussion of pragmatics and implicature: natural language users expect relevance and informativeness, not vacuous truths.

Programming analogies & vacuous truth

  • Several comments relate the puzzle to programming constructs:
    • Folding AND/OR over empty lists and the role of identity elements.
    • Functions like all/every returning true on empty collections.
  • Some see this as strong intuition for vacuous truth; others insist empty-set semantics should be explicitly modeled, especially when NULL/“no data” is meaningful.

Critiques of the puzzle and logic-puzzle culture

  • Some call the puzzle a “gotcha” or “midwit trap” that relies on an unspoken mapping from English to formal logic.
  • Others reply that this is precisely the point: to expose the gap between social intuition and formal reasoning.
  • There is broader debate about:
    • Whether such puzzles test reasoning or just familiarity with specific conventions.
    • How much weight to give formal logic when interpreting natural language.

Related topics and extensions

  • The thread branches into related paradoxes and puzzles (Monty Hall, ravens, liar paradox, two-envelope problems, “always lies” gods), used to illustrate similar tensions between intuitive and formal reasoning.