Optimal Paradigms — McCarthy 2005

[McC05] [Ken96]

Optimal Paradigms (OP) evaluates inflectional paradigms as candidates rather than individual forms: output-output faithfulness applies symmetrically across every member-pair, with no distinguished base. OP is McCarthy's successor to the Base-Identity / Uniform Exponence programme ([Ken96]), which first stated the symmetric all-pairs paradigm-uniformity constraint that Constraints.liftPairwise packages.

This file holds the OP substrate — the paper's framework apparatus, anchored to its originating paper and consumed by later study files. OP needs no new evaluation machinery: the existing tableau infrastructure is polymorphic in the candidate type C, so setting C := List Form gives paradigm-level evaluation for free. The OP-specific content is just the choice of constraint (mkOPMaxV) and the Majority Rules prediction (majority_minimizes_violations).

Main definitions

• mkOPMaxV — an OP-MAX-V constraint: symmetric pairwise OO-faithfulness on the vowel tier, built from Constraints.liftPairwise.
• majorityRules / majority_minimizes_violations — the Majority Rules prediction: an equivocal paradigm member aligns with the larger group.

Empirical status

[MR26] argue OP cannot simultaneously predict the distribution of schwa in verbs, nouns, and adjectives in Judeo-Tripolitanian Arabic — adjectives pattern phonologically with verbs but paradigmatically with nouns, contradicting OP's prediction that phonological behaviour tracks paradigm structure. See Studies/MarcoRasin2026.lean, which consumes mkOPMaxV from this file.

OP-MAX-V

def McCarthy2005.mkOPMaxV {Form : Type u_1} (vowelMismatch : Form → Form → ℕ) : Constraints.Constraint (List Form)

An OP-MAX-V constraint: the symmetric pairwise lift of OO-MAX restricted to the vocalic tier, summing a vowel-mismatch count over every paradigm member-pair. OP faithfulness is output-output (intra-paradigm); the mismatch callback supplies the tier-restricted comparison.

Equations

• McCarthy2005.mkOPMaxV vowelMismatch = Constraints.liftPairwise vowelMismatch

Majority Rules

def McCarthy2005.majorityRules (groupA groupB : ℕ) : Prop

The Majority Rules condition ([McC05]): when a paradigm member is equivocal between two forms (both satisfy higher-ranked markedness equally), OP predicts it aligns with the group containing more members.

majorityRules groupA groupB holds when group A outnumbers group B; under OP the equivocal member then surfaces with group A's form, because aligning with the larger group minimises total pairwise OP-MAX-V violations.

Equations

• McCarthy2005.majorityRules groupA groupB = (groupA > groupB)

@[implicit_reducible]

instance McCarthy2005.instDecidableMajorityRules (a b : ℕ) : Decidable (majorityRules a b)

Equations

• McCarthy2005.instDecidableMajorityRules a b = id inferInstance

theorem McCarthy2005.majority_minimizes_violations (a b : ℕ) (h : majorityRules a b) : (a + 1) * b < a * (b + 1)

When one group is larger, the equivocal member's OP-MAX-V violations are minimised by aligning with the majority. If a members have form X and b have form Y, choosing X gives (a + 1) * b cross-group pairs while choosing Y gives a * (b + 1); the former is smaller iff a > b.