12 / 18 · Think in objects and relations
Everyone has one, or one for everyone?
See why changing quantifier order changes who may depend on whom.
Builds on Not all is not none
Go to practice ↓A question to keep in mind
Every reviewer read a paper. Did all reviewers read the same paper?
Let x range over reviewers a, b and y over papers X, Y. R(x,y) means reviewer x read paper y. ∀x∃y R(x,y) permits a different paper for each reviewer. The choice of y may depend on x.
∃y∀x R(x,y) requires one paper read by every reviewer. A single y must work for all x. It entails the previous statement in these nonempty domains, but the converse can fail. Swapping two universal quantifiers or two existential quantifiers is different from swapping a mixed pair.
A free variable has not been bound by a quantifier. Its value must be supplied before a formula's truth is evaluated. 'Someone knows them' is not precise until the domain, the relation, and who 'them' denotes are fixed.
Work through an example
- a read only X; b read only Y. Each reviewer read something, so ∀x∃y R(x,y) is true.
- No column has every reviewer, so ∃y∀x R(x,y) is false. This is a finite countermodel to the converse implication.
- One shared paper would satisfy both claims, even if reviewers also read other papers.
Your turn
0 / 3Give each reviewer a paper, but leave no paper read by everyone.
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One possible world:
- a: R(a,X)=F, R(a,Y)=T
- b: R(b,X)=T, R(b,Y)=F
The diagonal works: a reads X, b reads Y, with no other links. The opposite diagonal works too.
Which implication is valid for these nonempty reviewer and paper domains?
Read solution · does not award completion
One shared paper implies each reviewer read a paper.
Keep track of whether the witness must remain fixed across all objects.
A study found a good prompt separately for each task. Which extra claim is not established?
Read solution · does not award completion
There is one fixed prompt that works for every task.
Quantifier order often hides inside informal claims about generalization.
Bring it back to your own work
Write a personal example of 'for everyone, something' and contrast it with 'something, for everyone'. Identify the shared witness that would be needed.
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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