11 / 18 · Think in objects and relations
Not all is not none
Negate quantified statements and recognize vacuous universals.
Builds on All, some, and finite worlds
Go to practice ↓A question to keep in mind
If no paper has been reviewed, is 'every reviewed paper is public' false?
¬∀x P(x) is equivalent to ∃x ¬P(x): not every object has P exactly when some object lacks P. ¬∃x P(x) is equivalent to ∀x ¬P(x): there is no witness for P exactly when all objects lack it.
The negation of ∀x(R(x) → C(x)) is ∃x(R(x) ∧ ¬C(x)). To break 'all red objects are circles', find a red non-circle. A gray square is not enough because it fails the restricting condition R.
If the domain contains no red objects, every instance of R(x) → C(x) is true. The universal is vacuously true; it does not assert that red objects exist. In modern first-order logic, add ∃x R(x) when existence is intended. Our domain stays nonempty even when its red subset is empty.
Work through an example
- Three files exist, and none has been reviewed. 'Every reviewed file is public' has no reviewed-and-private counterexample.
- The universal is true in this model, but 'a reviewed file exists' is false. This would be an unhelpful dashboard success signal unless review coverage is reported too.
- To negate 'not all files passed', remove the outer negation: all files passed. Do not replace it with 'no files passed'.
Your turn
0 / 3Make every red object a circle while making 'a red object exists' false.
Read solution · does not award completion
One possible world:
- a: red=F, circle=F
- b: red=F, circle=F
- c: red=F, circle=F
Remove redness from all three objects. The domain still contains a, b, c; only its red subset is empty.
Make 'every red object is a circle' false.
Read solution · does not award completion
One possible world:
- a: red=T, circle=F
- b: red=F, circle=F
- c: red=F, circle=F
At least one object must be red and not a circle. A single such object is sufficient.
What is the negation of 'Every sample passed'?
Read solution · does not award completion
At least one sample did not pass.
Negation excludes precisely the original claim, not more possibilities than necessary.
Bring it back to your own work
Find an 'all checks passed' message. Ask how many checks ran, whether any were skipped, and whether the tested set could be empty.
How do I retain independent judgment when using AI here?
Keep your own answer before consulting an AI candidate. Check sources, calculations, inference and conditions; explain adoption or rejection. Then solve a new-context task independently and apply the checks to this lesson.
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