Graf (2019): Monotonicity as an effective theory of morphosyntactic variation #
[Gra19] explains typological gaps — *ABA in adjectival gradation,
person-pronoun syncretism, case syncretism, noun stem allomorphy —
with two components: a fixed base hierarchy of prominence relations,
and the requirement that the form assignment from it be feasibly
monotonic (Morphology.Containment.FeasiblyMonotone, his def. (6)).
The keystone here is isContiguous_tfae: contiguity, feasible
monotonicity, Elsewhere-generability, and Superset-spellability
coincide over linear hierarchies — Graf's "effective theory" and the
two insertion mechanisms are provably one constraint. Graf establishes
the 3-cell case by exhaustion and verifies larger hierarchies
instance-by-instance; the general equivalence is
Morphology.Containment.isContiguous_iff_feasiblyMonotone, which this
file instantiates on his Table 1 (adjectival gradation, via the
English fragment) and Table 2 (person-pronoun syncretism,
[Har15]'s survey).
Main declarations #
isContiguous_tfae— contiguous ↔ feasibly monotonic ↔ Elsewhere-generable ↔ Superset-spellableenglish_suppletion_feasiblyMonotone,person*— Tables 1–2 instantiatedaab_feasiblyMonotone,aab_not_groundedly_realizable— the AAB division of labor
TODO #
Graf's preferred case-layer hierarchy, the PCC person-pair poset, and
the GCC gender poset are genuinely partial orders — and the PCC/GCC
maps go into the fixed two-element truth-value algebra (upper sets,
not kernels) — so they need a partial-order generalization of
Pattern and an upper-set substrate. Concrete deferred targets: his
ban (14) on multiple cross-level case syncretisms; McFadden's
nominative stem-allomorphy generalization (16) via the conflated
hierarchy's cycle-forces-identity argument; the exactly-nine monotone
PCC maps as a finite check.
The keystone: four characterizations coincide #
Over a linear containment hierarchy the following are equivalent: the pattern is contiguous; it is feasibly monotonic ([Gra19] def. (6)); it is generable by Elsewhere insertion over a terminal antihomophonous vocabulary (DM, [Bob12] ch. 2); it is spellable by Superset competition over a context-free antihomophonous lexicon (nanosyntax, [Cah09]). Graf's "effective theory" and the two insertion mechanisms are one constraint.
Table 1: adjectival gradation #
Every English fragment suppletion pattern is feasibly monotonic — [Gra19] Table 1's attested rows (AAA smart, ABB good), run over the whole fragment.
The unattested *ABA row: no codomain order makes good – better – goodest monotone ("Irrespective of how this set is ordered, there can
be no monotonic function f with f(1) = f(3) ≠ f(2)").
Table 2: person-pronoun syncretism #
[Gra19] §3.1 runs the same hierarchy shape over person,
1 > 2 > 3 (the Zwicky hierarchy, [Zwi77]; the repo's
feature-level anchor for it is Person.hierarchyRank), against
[Har15]'s pronoun-inventory survey (Graf's Table 2): attested
partitions include 1|2|3 (Jarawa, Kiowa), 1|23 (Damin), and crucially
12|3 (Winnebago) — an AAB shape — while 13|2 is missing from the
survey, as monotonicity predicts: "the account provides a perfect fit
for the typology [of pronouns], whereas it only carves out a superset
of the attested patterns for adjectival gradation and case
syncretism." The encoding runs the hierarchy with index ascending
1 → 3, the dual of Graf's prominence order — equivalent for (feasible)
monotonicity, his fn. 1.
Jarawa, Kiowa: all three persons distinct (1|2|3).
Equations
- Graf2019.personJarawa = ![0, 1, 2]
Instances For
Damin: second and third person share a form (1|23).
Equations
- Graf2019.personDamin = ![0, 1, 1]
Instances For
Winnebago: first and second person share a form (12|3) — an AAB shape, attested in person though not in gradation.
Equations
- Graf2019.personWinnebago = ![0, 0, 1]
Instances For
The unattested partition 13|2: first and third persons syncretic to the exclusion of second.
Equations
- Graf2019.personGap = ![0, 1, 0]
Instances For
The attested person partitions are all feasibly monotonic over
1 > 2 > 3.
The missing partition is exactly the non-monotonic one.
The AAB division of labor #
[Gra19]: "monotonicity cannot give a unified explanation of the
absence of both ABA and AAB patterns. However, this is actually a
welcome state of affairs because AAB patterns do show up in other
empirical domains" — attested in person (Winnebago), unattested in
adjectival gradation, where the absence "has to be stipulated", e.g.
in terms of syntactic containment (his suggestion). The pair below
states the division precisely: AAB is feasibly monotonic, and its
gradation-side exclusion is carried by realization-level conditions
([Bob12]'s markedness condition (202), via
Morphology.Containment.csg2) — which the person attestation shows
are not category-general, the same point the pronominal case/number
data make in Studies/SmithMoskalEtAl2019.lean.
AAB is feasibly monotonic: Graf's account cannot — and, given Winnebago, should not — exclude it.
Under antihomophony plus the markedness condition (202), no vocabulary realizes an AAB pattern — the gradation-side exclusion, carried by the realization engine rather than by monotonicity.