How to Study Boundary Phenomena

What makes a category real?

Traditional answer: necessary and sufficient conditions
Problem: linguistic categories resist definition. Pronouns share properties, but no single property is shared by all and only pronouns.
Alternative: a robust cluster of co-occurring properties + stabilizing mechanisms that maintain the cluster = a real category

Homeostatic = self-correcting, like body temperature: the system drifts, mechanisms push it back. A category maintained this way is projectible – you can make predictions about new instances.

~2 min. Open with the problem that motivates the whole talk. Traditional categories need necessary and sufficient conditions, but grammatical categories don’t have those. HPC theory comes from philosophy of biology (Boyd 1991, 1999) – it’s how biologists think about species. Miller (2021) brought it to linguistics. The name: homeostatic means self-correcting, like body temperature – the system has mechanisms that push it back when it drifts. Property cluster = the category is defined by a cluster of co-occurring properties, not by necessary and sufficient conditions. If the audience looks blank at homeostatic, the body-temperature analogy lands well. The spinning top slide will reinforce this. Key term to land: projectible – you can make predictions about new instances. Delivery tip (Harris): consider opening with the particular (each other) before the general (HPC theory) – the theory may land better after the puzzle is on the table. Even a one-sentence teaser works.

No real kind without a purpose

Syntactician’s proper noun

Distribution: Brett left / *The Brett left
Agreement: 3rd person singular
Modification: restricted
Typically has proper name semantics

Semanticist’s proper name

Rigid designation
Referential opacity
Sense vs. reference
Usually instantiated by a proper noun

Brett is both. Different projections, different mechanisms, different HPCs, same extension.

Homeostasis: the virtuous circle

What holds these clusters together?

Property cluster
co-occurring properties

stabilizes →

← sustains

Mechanisms
causal processes

Five grammatical mechanisms: morphological realization rules · agreement/case systems · entrenched distributional patterns · grammaticalization pathways · community norms

Mechanisms maintain clusters. Clusters maintain mechanisms. That’s what homeostatic means. (A reciprocal relationship, as it happens.)

Stability is dynamic, not static

Grammatical categories are spinning tops, not balls in valleys.

The reciprocals puzzle

Our question is syntactic: what lexical category do each other and one another belong to? Pronoun or compound determinative?

Determinative: the word class of the, some, every — not the determiner function. Compound determinatives like somebody (some + body) and everyone (every + one) share the compound structure of each other (each + other).

	Pronoun-like	Determinative-like
Morphology (66)		Compound; not monomorphemic; no distinct accusative, genitive, or reflexive forms
Semantics (36)	Pro-form (stands in for a full NP); definite; anaphoric; requires an antecedent
Syntax (50)	Not in partitives (*each other of the people); not in existentials; doesn’t take else	Accepts genitive ’s; appears in object
Phonology (3)		(weak signal)

155 properties. Morphology pulls one way, semantics the other, syntax is mixed. Which way do they go?

~2 min. Start by landing the pivot: we’re asking a syntactic question, so morphological and semantic facts are evidence, not the answer itself. Pullum’s keynote is about category/function confusions, so this audience will appreciate the precision. Quick terminology: determinative is a word class (category), determiner is a function — the distinction Pullum is keynoting on. Compound determinatives like somebody, everyone, nothing have compound morphology and serve as fused determiner–head in NP: one word does both jobs. Reciprocals have the same compound structure (each + other), which looks determinative-like. But semantically they’re pro-forms, definite, anaphoric — they require an antecedent that structurally outranks them, like pronouns. Morphologically they lack the distinct accusative, genitive, and reflexive forms that pronouns like he/him/his/himself have — more like compound determinatives. Syntactically it’s mixed: they don’t function as fused determiner–head (pronoun-like), and they don’t appear in existentials or take else (pronoun-like), but they accept genitive ’s and appear in object position (like somebody). The numbers in parentheses show how many features per block — these are just highlights from 155 total. Agreement is 3rd person either way, so it’s not diagnostic. This is the kind of problem where cherry-picking is easy and tempting.

The problem with cherry-picking

Two items, 155 tests, and a strong temptation to cherry-pick.

Croft (2001) calls this methodological opportunism: consciously or not, we select tests that support our preferred analysis.

The alternative: measure the stability of diagnostic ambiguity. Vary every reasonable analytic choice and ask whether the answer changes.

The interesting question isn’t “which category?” but “how stable is the apparent boundary position under different measurement choices?”

What HPC predicts for boundary items

Stable position: the result doesn’t depend on how you measure
Cross-dimensional tension: morphology and semantics pull in different directions
Clean anchors: clear cases come out right, so the method is trustworthy
Near-parity mixture: the item sits right at the midpoint between the two categories
Robustness to null: scramble the data keeping its basic structure; the pattern shouldn’t appear by chance — and it doesn’t

These aren’t arbitrary desiderata. They’re consequences of the theory.

Mapping grammatical space

155 binary properties (morphology, syntax, semantics, phonology) across 138 items. This 2D projection captures ~17% of the variance; all actual measurement uses full 155-dimensional Jaccard distances.

Multiple Correspondence Analysis projection. Pronouns (blue) and determinatives (red) form regions; compound determinatives sit at the interface; reciprocals (triangles) fall in that interface zone.

Not a statistical fluke

Scramble the data 5,000 times, preserving how many properties each word has and how many words have each property. This tests whether the specific combination of features drives reciprocals’ position, not just marginal structure.

Observed pattern in only 0.6% of scrambles (p = 0.006).

Permutation null distribution. Dashed line marks the observed value.

Stable across analytic choices

Vary every reasonable analytic choice (distance metric, which properties, weighting) and show all results. Each point is one specification; Delta = mean distance to pronouns minus mean distance to determinatives.

Four metrics (Jaccard, Dice, Hamming, IDF-weighted) × two regularizations (ridge, elastic net). Positive = closer to determinatives; negative = closer to pronouns. Exception: removing morphology flips the sign.

Sign stable across most choices. Removing morphology flips it. That’s cross-dimensional tension.

Right at the midpoint

Best-fitting mixture weight: each other = 0.534, one another = 0.487

Every item sorted from determinative (0) to pronoun (1). Reciprocals sit at the midpoint.

All five expectations confirmed

	Expectation	Result
✔	Stable position	Result stable no matter how you measure
✔	Cross-dimensional tension	Morphology → determinative; semantics → pronoun
✔	Clean anchors	Same methods correctly identify clear cases
✔	Near-parity mixture	Best-fitting weights 0.534, 0.487 (near midpoint)
✔	Robustness to null	Pattern in only 0.6% of scrambled data

This isn’t measurement failure. It’s what a real boundary looks like.

What kind of problem is this?

Reciprocals are one or the other. But our instruments can’t resolve which.

Resolved Unresolved

Categories are internally gradient but sharply bounded. This isn’t gradience; it’s a boundary phenomenon: independent mechanisms sustaining opposed pulls.

~1.5 min. The slide has two beats: the diffraction image, then the thesis sentence. Open with the image: reciprocals are one category or the other, but our diagnostics can’t resolve which – just like two points below the diffraction limit. Let the image sit for a moment before clicking through. Then land the thesis: this isn’t gradience (smooth fading of membership across a single dimension). Gradience predicts uniform weakening across all features. What we see is the opposite – morphology pulls one way, semantics the other, each stably. That’s the signature of distinct mechanisms maintaining overlapping bundles. The boundary is ontologically sharp but epistemically inaccessible. Spare material for Q&A: (1) Peirce’s scholastic realism – categories are real, not convenient fictions; fallibilism – our knowledge of them is always provisional. (2) The empirical pattern is compatible with Slater’s Stable Property Clusters and Khalidi’s causal-network account. What distinguishes HPC is the emphasis on cross-dimensional tension as the signature of distinct mechanisms. (3) If boundaries were sharp but arbitrarily located, we’d expect greater sensitivity to metric choice than the specification curve reveals.

Back to the spinning top

Property cluster
co-occurring properties

stabilizes →

← sustains

Mechanisms
causal processes

Determinative region (morphology pull)

Morphological realization rules maintain compound structure: each + other

Pronoun region (semantics pull)

Interpretive mechanisms (reference tracking, anaphoric dependencies) maintain pronoun-like behaviour

Two sets of mechanisms, two spinning tops, one boundary. Stop either spin and the tension dissolves.

~1.5 min. Bring it full circle. The virtuous circle from slide 4 now has content: the determinative region is maintained by morphological realization rules (compound structure), the pronoun region by interpretive mechanisms (reference tracking, binding). Reciprocals sit where these two sets of mechanisms overlap and conflict. That’s why the cross-dimensional tension exists and why it’s stable. If asked about the stabilizes/sustains dynamic, use the footpath analogy. A footpath across a field: the cluster is the observable bundle of properties (worn ground, short grass, compacted soil, drainage follows it). The mechanism is repeated foot traffic. Foot traffic produces and maintains all those properties as a bundle. The visible path (the cluster) attracts more walkers (sustains the mechanism). Take away the walkers and the properties stop co-occurring: grass grows back, soil loosens, drainage shifts. They were bundled because something kept them bundled. For reciprocals: the pronoun-like cluster is the observable bundle (anaphoric, requires an antecedent, definite, pro-form). The mechanism is reference tracking in the grammar. Reference tracking keeps those properties bundled. Because the bundle exists as a stable, learnable pattern, new speakers acquire it, which perpetuates the reference-tracking behaviour. The cluster is what you can see. The mechanism is why it holds together. The loop is that what holds together is what gets learned.

How to study boundary phenomena

Build comprehensive profiles (don’t cherry-pick diagnostics)
Test against scrambled baselines (especially with small n)
Vary specifications systematically (show all results)
Calibrate against clear cases (verify known structure)
Ask whether the ambiguity is stable

We asked what makes a category real. The HPC answer: maintenance. Stable ambiguity is evidence of that maintenance at work.

Paper: LingBuzz 009294 · Code: GitHub · brettreynolds.ca · brett.reynolds@humber.ca