Foundation B · beneath the house
The Rule Garden
“Can a small rule grow bigger than its parent?”
Lamp waiting
▶ Hear the storyteller tell this roomLook
A small garden with branching paths. A sign at the gate says which flowers are allowed.
Try
Put up signs on the smaller paths. See which ones hold.
Notice
Try the experiment first. Then this will fill in.
What keeps this room working?
A child rule can only keep what its parent already allows.
What would break this room?
If a child rule could create a permission its parent never had.
Child-sized version
“You can make a rule smaller. You cannot secretly make it bigger.”
Try to break the room
For grown-upsShow me the math
Rules are sets of allowed possibilities, ordered by inclusion. Narrowing moves down the order. Combining two rules takes their meet: the possibilities both allow. No top-most master rule is needed for this to work.
- Constraint set
- The set of possibilities a rule allows.
- Partially ordered set
- Rules ordered by 'is at least as strict as'.
- Meet-semilattice
- Any two rules have a greatest common narrowing: keep what both allow.
- Composition by overlap
- Rules combine where they agree. No single rule must contain everything.