How to Study Boundary Phenomena

The puzzle

Each other and one another

Everyone calls them pronouns.

But how do we know that’s the right lexical category?

And when there’s only two of them, …?

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 a Watt governor: 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: bare NP position
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

sustains →

← stabilizes

Mechanisms
causal processes

Mechanisms maintaining grammatical categories:

Acquisition – children converge on categories from distributional input
Entrenchment – high-frequency items anchor the cluster
Interactive alignment – speakers converge in conversation
Iterated transmission – learnable structure survives across generations
Functional pressure – categories persist because they’re useful

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

~1.5 min. The core theoretical machinery. Bridge from slide 3: we saw that different mechanisms produce different kinds. Now: what’s the relationship between mechanisms and properties? Properties cluster because mechanisms maintain them. But the cluster itself sustains the mechanisms (speakers learn and reinforce the patterns because the cluster exists). This reciprocal reinforcement is what makes HPC categories real and stable. That’s the “homeostatic” in HPC – self-correcting, like the Watt governor on slide 2. The five mechanisms are from the book (Ch. 4), following the biological parallel: acquisition = transmission, entrenchment = frequency anchoring, alignment = cohesion, transmission = filtering, functional pressure = attraction. We’ll come back to this diagram at the end when we see which mechanisms are pulling reciprocals in which direction. Q&A note: if someone asks about morphological rules or agreement systems, those are products of these mechanisms, not mechanisms themselves – they’re what gets maintained, not what does the maintaining.

Stability is dynamic, not static

Grammatical categories are spinning tops, not balls in valleys.

The data

I gathered every property I could think of, however trivial, and coded them for all the CGEL pronouns (65) and determinatives (73).

Property	each other	one another	they	somebody
Monomorphemic			Y
Definite	Y	Y	Y
Anaphoric	Y	Y	Y	Y
Fused determiner-head				Y
Appears in object	Y	Y		Y
Requires antecedent	Y	Y

155 binary properties × 138 items. The goal: leave no room for cherry-picking.

The reciprocals puzzle

	Pronoun-like	Determinative-like
Morphology (66)		Compound; no distinct accusative, genitive, or reflexive forms
Semantics (36)	Definite; anaphoric; requires an antecedent
Syntax (50)	Not in partitives; not in existentials; no else	Accepts ’s; appears in object
Phonology (3)		(weak signal)

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

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.

Each point is one analytic specification: different distance metrics, different feature weightings. Positive = closer to determinatives; negative = closer to pronouns.

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

Right at the midpoint

Best-fitting mixture weight: each other ~0.5, one another ~0.5. Remove morphology: both jump to ~0.94 (strongly pronoun-like). Remove semantics: both shift toward determinative (Delta ~ +0.09). The midpoint exists because morphology and semantics are pulling in opposite directions.

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

~1 min. The mixture calibration asks: if reciprocals are a blend of the two anchor profiles, what blend best predicts their observed properties? Answer: almost exactly 50/50. These mixture weights (0.534, 0.487) are SSE-minimizing blends from the calibration curve. Separately, a classifier assigns near-chance pronoun probabilities (0.485 for each other, 0.467 for one another), and the predictive log-likelihood is identical under either labelling (-54.240). Two independent methods, same verdict: the model genuinely can’t tell. Both weights are stable under modest feature ablations and alternative scoring rules (spec curve, previous slide). Q&A: if asked about credible intervals, the sim-recovery calibration shows that at w=0.50 the empirical values fall at the 58th and 47th percentiles of simulated data – well within the expected range under a true 50/50 mixture. The point estimates lack formal CrIs because they come from a calibration curve, not a posterior, but the sim-recovery analysis provides the equivalent uncertainty quantification.

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 ~0.53, ~0.49 (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 – and it tells you what you can predict: roughly half a pronoun’s behaviour, half a determinative’s.

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.

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
Cash out the projective consequences (what does the classification predict?)

Categories are real because they’re projectible. Maintenance is the mechanism; projection is the payoff. Stable ambiguity tells you exactly how much projection each anchor category provides.

Paper: LingBuzz 009294 · Code: GitHub · R package in progress · brettreynolds.ca · brett.reynolds@humber.ca