Negative Attributes via Proper Coarsening

[Mer99]

Epistemic, syntax-independent characterization of negativity:

• isProperCoarsening q q' elements — Q coarsens Q' and has strictly fewer cells over elements.
• isNegativeAttribute R q elements — binaryPartition R is a proper coarsening of q.

[Mer99]: a predicate is negative not because of morphological form (un-, not, etc.) but because its yes/no distinction is strictly coarser than the partition under discussion — answering "does R hold?" discards information q distinguishes.

def QUD.isProperCoarsening {M : Type u_1} [DecidableEq M] (q q' : QUD M) (elements : List M) : Prop

Q is a proper coarsening of Q' over a finite domain iff Q coarsens Q' and has strictly fewer cells.

[Mer99] defines negative attributes via proper coarsening: R is a negative attribute with respect to partition F iff for some Q ∈ F, {R, Q} is a proper coarsening of F. This characterization is purely epistemic and syntax-independent — negativity is a matter of partition kinetics, not morphological form.

Equations

• q.isProperCoarsening q' elements = (q.coarsens q' ∧ q.numCells elements < q'.numCells elements)

def QUD.isNegativeAttribute {M : Type u_1} [DecidableEq M] (R : M → Bool) (q : QUD M) (elements : List M) : Prop

A predicate R is a negative attribute with respect to partition Q over a finite domain iff the binary partition of R is a proper coarsening of Q.

[Mer99]: negativity is not a syntactic property (presence of "un-", "not", etc.) but a partition-kinetic one. R is negative relative to Q when the R/¬R distinction is strictly coarser than Q's partition — answering whether R holds loses information that Q distinguishes.

Equations

• QUD.isNegativeAttribute R q elements = (QUD.binaryPartition R).isProperCoarsening q elements