A related perspective on meagre sets as propositions (mostly writing down for my own interest):
The interior operator IntAcan be thought of as “rounding down to the nearest observable proposition”, since it is the upper bound of all opens that imply A.
The condition for A to be nowhere dense is equivalent to Int¬Int¬A=∅.
If we are working with a logic of observables, where every proposition must be an observable, the closest we can get to a negation operator is a pseudo-negation ∼:=Int¬.
So a nowhere dense set is a predicate whose double-pseudo-negation ∼∼A is false, or equivalently ∼∼∼A is true.
Another slogan, derived from this, is “a nowhere dense hypothesis is one we cannot rule out ruling out”.
The meagre propositions are the σ-ideal generated by nowhere dense propositions.
A related perspective on meagre sets as propositions (mostly writing down for my own interest):
The interior operator IntAcan be thought of as “rounding down to the nearest observable proposition”, since it is the upper bound of all opens that imply A.
The condition for A to be nowhere dense is equivalent to Int¬Int¬A=∅.
If we are working with a logic of observables, where every proposition must be an observable, the closest we can get to a negation operator is a pseudo-negation ∼:=Int¬.
So a nowhere dense set is a predicate whose double-pseudo-negation ∼∼A is false, or equivalently ∼∼∼A is true.
Another slogan, derived from this, is “a nowhere dense hypothesis is one we cannot rule out ruling out”.
The meagre propositions are the σ-ideal generated by nowhere dense propositions.