Projecting 364 legal terms onto six axes drawn from doctrine and political theory.
Experiment §3.2 — Value axes
The second experiment projects the 364 legal terms onto six axes
drawn from legal doctrine and political theory:
individual ↔ collective, rights ↔ duties,
public ↔ private, state ↔ market,
natural ↔ positive, status ↔ contract. Each axis is
built from ten antonym pairs (Kozlowski, Taddy & Evans, 2019) in
each language — English on the Western-trained side, Chinese on the
Chinese-trained side. The axis is the L2-normalised mean of the ten
pole-difference vectors; each term receives a signed score on each
axis as its cosine with the axis direction.
The section answers four nested questions. Are the axes coherent —
do the ten antonym pairs that built each axis project on the side
they were chosen to represent (§3.2.1)? Are the six axes independent
directions in the embedding space, or do they partly overlap
(§3.2.2)? When two models score the same 364 terms on the same axis,
do their rankings agree more within a tradition than across (§3.2.3,
§3.2.4)? And on which axes do the two traditions diverge most
(§3.2.4, §3.2.5)?
§3.2.1 · Building an axis from pairs of opposites
Legal scenarioLegal doctrine treats contracts as the antithesis of status, rights as the antithesis of duties. If a language model encodes legal vocabulary, the ten antonym pairs that define each axis should mostly project on the side they were chosen to represent.
Result in wordsOn the six axes and six monolingual models, the great majority of cells return ten out of ten (or nine out of ten) antonym pairs aligned with their nominal pole; no cell falls below seven of ten. The few soft cells flag axes where one specific pole word sits closer to its opposite under the contextualisation of Hong Kong ordinances — informative about doctrinal placement, not about axis formation.
Sanity heatmap: ratio of antonym pairs aligned with their nominal pole, one cell per axis × model (attested encoding). The ten English and ten Chinese antonym pairs that build each axis are listed verbatim under Inside the inputs.
Take-homeAxis construction is sound on the majority of cells. The few low-pass cells are the early warning of the pool-sensitivity reported in §3.2.4.
axes = 6 · antonym pairs / axis = 10 · languages = English + Chinese · models = 5 EN + 5 ZH
Pass means at least half the ten pairs project on the side they were chosen for. The averaging step follows Kozlowski, Taddy & Evans (2019).
§3.2.2 · Axes independence
Legal scenarioPublic ↔ private and state ↔ market sound related; rights ↔ duties and individual ↔ collective do too. If two axes occupied the same direction in embedding space, projecting the 364 terms on both would be redundant. How much do the six axes share?
Result in wordsOn the representative reading (BGE-EN-large, attested), the fifteen off-diagonal cosines range between −0.21 and +0.34 in signed value, with mean magnitude 0.13. No pair is collinear; no pair is exactly orthogonal. Individual / collective and rights / duties are the most aligned (cos ≈ +0.34, the “individual” pole near the “rights” pole); natural / positive is the closest of the six to a direction independent of the others.
Inter-axis cosine matrix, attested encoding. The diagonal is unity by construction; off-diagonal values are signed cosines. Use the dropdown to switch model — the qualitative shape of the matrix recurs across traditions, the magnitudes vary.
Take-homeSix axes occupy six distinct directions: the per-axis readings of §3.2.4 are not six restatements of the same measurement. Doctrinal proximity (individual / rights, state / status, public / state) shows up as moderate but bounded alignment, never as collinearity.
Off-diagonal range on the representative model: [−0.21, +0.34], mean magnitude 0.13. The full per-model matrices are referenced from §3.2.2 of the thesis.
§3.2.3 · Agreement on the ranking
Legal scenarioEach model assigns a signed score to each of the 364 terms on each of the six axes. When two models are compared on the same axis, their rankings of the 364 terms should correlate strongly within a tradition and more weakly across — if the axes track tradition-specific legal meaning.
Result in wordsDistribution of Spearman ρ across 45 pairs per axis, attested encoding. Within-tradition rankings agree strongly on most axes; cross-tradition rankings agree more weakly — and the spread varies systematically by axis. The §3.1.3 cohort-level pattern transfers to the per-axis level.
Box plot of per-pair ρ for each axis, attested. Points are individual model pairs, coloured by group.
Take-homePer-pair ρ distributions are tight within tradition and broader across. The single-axis projection picks up the same structural agreement that §3.1.3 reads on the full distance map.
Technical apparatusOpen technical detail
pairs total = 270 · pairs / axis = 45 · axes = 6 · metric = Spearman ρ on the 364-term rank vector
Each per-pair entry stores a 95% confidence interval from term-level block bootstrap (B = 10 000).
§3.2.4 · Cross-linguistic agreement — which axes diverge most?
Legal scenarioAmong the six axes, which is the least shared across traditions — the one on which Western-trained and Chinese-trained models rank the 364 terms most differently? And by which margin? Doctrine has its own intuitions: natural-versus-positive law, state-versus-market allocation, individual-versus-collective normative weight all carry tradition-specific framing.
Result in wordsOn attested encodings, the most divergent axis is natural ↔ positive with cross-tradition mean ρ = 0.092. The least divergent is rights ↔ duties with mean ρ = 0.394. The ranking is curation-sensitive on three of the six axes: individual / collective, public / private and natural / positive hold their cross-tradition mean under pool perturbation, while rights / duties, status / contract and state / market shift substantially when the curated pool shifts. The pool sensitivity is itself the methodological reading of §4.2 of the thesis.
Cross-tradition mean ρ per axis, most divergent on top. Use the toggle above the chart to switch between attested (default) and bare encoding — five of the six axes diverge more under attestation; rights / duties is the only axis on which the two encodings coincide.
The same data, bare and attested side by side. The exception (rights / duties) is visible as the axis where the two bars overlap.
Axis
Cross-tradition ρ̄ (attested)
natural ↔ positive
0.092
state ↔ market
0.125
individual ↔ collective
0.186
public ↔ private
0.288
status ↔ contract
0.363
rights ↔ duties
0.394
Pool-sensitivity warning. On rights / duties, status / contract and state / market the cross-tradition ρ̄ shifts by more than 0.05 under realistic pool perturbations, so the rank order of these three axes is not invariant. On individual / collective, public / private and natural / positive the ρ̄ is stable under the same perturbations. Read the ranking as an ordering within an internally-scaled column, not as a cross-axis comparison of magnitudes (§4.2).
Take-homeThree axes carry a stable cross-tradition signature, three are sensitive to curation. The reading is ordinal — the ranking is the substantive measurement — not metric: rights / duties at 0.394 is not “twice as agreed” as individual / collective at 0.186.
Technical apparatusOpen technical detail
most divergent = natural_positive · 0.092 · least divergent = rights_duties · 0.394 · axes total = 6 · cross pairs / axis = 9 monolingual (3 EN × 3 ZH)
Each cell is the mean Spearman ρ between the 364-term rankings of one English-side model and one Chinese-side model on the same axis.
Legal scenarioBeyond an axis-level ρ, the substantive question is: which individual terms anchor the cross-tradition divergence? Their identity tells a doctrinal story that the aggregate ρ cannot.
Result in wordsOn natural / positive, the most divergent terms are prejudice, discrimination, punishment, religion, perjury: the Western-trained reading sends them toward the “natural” pole (the offence is wrong before being criminalised); the Chinese-trained reading sends the same five toward the “positive” pole (the offence is the statutory text that criminalises). On individual / collective, the divergence reverses polarity: terms of bilateral private-law relation (compensation, obligation, counterparty) sit on the individual side for the Western-trained models and on the collective side for the Chinese-trained ones. On rights / duties, both readings of the term freedom / 自由 collapse to the same Chinese lemma but project a quarter-axis apart (|Δ| ≈ 0.30). Use the axis dropdown to browse the top ten divergent terms on each axis.
Per-axis top-ten cross-tradition divergent terms, sorted by |Δ| from largest at the top. Blue bar: mean projection of the term across the three Western-trained models. Red bar: mean projection across the three Chinese-trained models. |Δ| is in the hover. Use the axis dropdown to switch between the six axes (attested encoding).
Take-homeTerm-level and axis-level divergence converge: where the axis ρ̄ is lowest, the top divergent terms project on opposite poles; where the axis ρ̄ is highest, the divergence concentrates on a single term (freedom / 自由). §3.2.5 of the thesis is the exhibit catalogue.
|Δ| values are not comparable across axes: each axis carries its own projection scale. The intra-axis ranking is meaningful; the cross-axis magnitude comparison is not (§4.2).