Gärtner & Gyuris (2017): Delimiting the Space of Bias Profiles #
@cite{gartner-gyuris-2017}
Formalization of the bias profile framework from @cite{gartner-gyuris-2017}, which defines bias profiles as non-empty power-set choices from {+, −, %} for evidential and epistemic dimensions across PPQ/IN-NPQ/ON-NPQ forms.
Key Results #
- Space size: 7³ × 7³ = 117649 total profiles
- 7 delimiting principles (§2) reduce the space progressively:
- No Uniformity (§2.1)
- PPQ ≠ NPQ (§2.2)
- Markedness (§2.3)
- Polarity Match / QA Alignment (§2.4)
- Convexity (§2.5)
- Narrow Epistemic Choice (§2.6)
- Static Complementarity (§2.7)
- Final reduction: Convexity + Narrow Epistemic Choice + Static Complementarity together yield (4 × 2)³ = 512 permissible profiles.
Cross-Linguistic Data (Appendix A) #
Six bias profiles from English, Japanese, and Hungarian are encoded and verified against the delimiting principles.
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A bias choice is a non-empty subset of {+, −, %}. There are 2³ − 1 = 7 such subsets. We represent them as sorted lists for decidable equality and enumeration.
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The 7 non-empty subsets of {+, −, %}.
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Bias dimension: evidential vs epistemic.
- evidential : BiasDimension
- epistemic : BiasDimension
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Map G&G forms to Romero's typology.
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- Phenomena.Questions.Studies.GartnerGyuris2017.GGPQForm.PPQ.toRomero = Semantics.Modality.BiasedPQ.PQForm.PosQ
- Phenomena.Questions.Studies.GartnerGyuris2017.GGPQForm.IN_NPQ.toRomero = Semantics.Modality.BiasedPQ.PQForm.LoNQ
- Phenomena.Questions.Studies.GartnerGyuris2017.GGPQForm.ON_NPQ.toRomero = Semantics.Modality.BiasedPQ.PQForm.HiNQ
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A cell in the bias profile grid: one PQ form × one bias dimension.
- form : GGPQForm
- dim : BiasDimension
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A bias profile assigns a non-empty bias choice to each of 6 cells (3 PQ forms × 2 bias dimensions).
- ppqEv : BiasChoice
- ppqEp : BiasChoice
- inNpqEv : BiasChoice
- inNpqEp : BiasChoice
- onNpqEv : BiasChoice
- onNpqEp : BiasChoice
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- Phenomena.Questions.Studies.GartnerGyuris2017.instBEqBiasProfile.beq x✝¹ x✝ = false
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Access a bias profile by cell.
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- bp.get { form := Phenomena.Questions.Studies.GartnerGyuris2017.GGPQForm.PPQ, dim := Phenomena.Questions.Studies.GartnerGyuris2017.BiasDimension.evidential } = bp.ppqEv
- bp.get { form := Phenomena.Questions.Studies.GartnerGyuris2017.GGPQForm.PPQ, dim := Phenomena.Questions.Studies.GartnerGyuris2017.BiasDimension.epistemic } = bp.ppqEp
- bp.get { form := Phenomena.Questions.Studies.GartnerGyuris2017.GGPQForm.IN_NPQ, dim := Phenomena.Questions.Studies.GartnerGyuris2017.BiasDimension.evidential } = bp.inNpqEv
- bp.get { form := Phenomena.Questions.Studies.GartnerGyuris2017.GGPQForm.IN_NPQ, dim := Phenomena.Questions.Studies.GartnerGyuris2017.BiasDimension.epistemic } = bp.inNpqEp
- bp.get { form := Phenomena.Questions.Studies.GartnerGyuris2017.GGPQForm.ON_NPQ, dim := Phenomena.Questions.Studies.GartnerGyuris2017.BiasDimension.evidential } = bp.onNpqEv
- bp.get { form := Phenomena.Questions.Studies.GartnerGyuris2017.GGPQForm.ON_NPQ, dim := Phenomena.Questions.Studies.GartnerGyuris2017.BiasDimension.epistemic } = bp.onNpqEp
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Total space: 7 choices per cell, 6 cells = 7⁶ = 117649.
No Uniformity: a bias profile is not entirely uniform, i.e., not all 6 cells have exactly the same bias choice.
"none of them consist of exactly the same choice, e.g., {+}, for each of its 6 dimensions."
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No Uniformity removes exactly 7 profiles (one per uniform choice).
PPQ ≠ NPQ: Negation has an impact on bias. Both the evidential AND epistemic choices of PPQ must differ from those of each NPQ form.
"PPQ ≠ NPQ (interpreted more precisely as PPQ^ev ≠ NPQ^ev & PPQ^ep ≠ NPQ^ep)" — §2.2.
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Quantitative Markedness (distributive, §2.3 eq. 11a): expressing marked (negative) meanings does not lead to more options than expressing their unmarked (positive) counterpart.
|PPQ^ev| ≥ |IN-NPQ^ev| & |PPQ^ev| ≥ |ON-NPQ^ev| & |PPQ^ep| ≥ |IN-NPQ^ep| & |PPQ^ep| ≥ |ON-NPQ^ep|
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Quantitative Markedness (collective, §2.3 eq. 11b): |PPQ^ev| + |PPQ^ep| ≥ |IN-NPQ^ev| + |IN-NPQ^ep| & |PPQ^ev| + |PPQ^ep| ≥ |ON-NPQ^ev| + |ON-NPQ^ep|
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Distributive Markedness yields 33856 profiles (Appendix B [1]). Per dimension: 3×3² + 3×6² + 1×7² = 184 (PPQ,NPQ) triples. Two independent dimensions: 184² = 33856.
Collective Markedness yields 56536 profiles (Appendix B [2]): 9×9² + 18×27² + 15×42² + 6×48² + 1×49² = 56536, summed over |PPQ^ev|+|PPQ^ep| = 2..6, counting NPQ pairs with sum ≤ PPQ sum.
Avoid Disagreement: − ∉ PPQ and + ∉ NPQ. The polarity of the question and the direction of bias should not totally contradict each other.
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Don't Rule Out Agreement: each cell of PPQ must contain +, and each cell of NPQ must contain −. The constraint applies per-cell, not per-row.
This yields 4 choices per cell (subsets containing the matching polarity), so 4⁶ = 4096 total (§2.4, chart (19)).
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Avoid Disagreement yields 3⁶ = 729 profiles.
Don't Rule Out Agreement yields 4⁶ = 4096 profiles.
A bias choice is convex if it doesn't "skip" intermediate values in the Hasse ordering + > % > −. Concretely, {+, −} is ruled out because it crosses over % without including it.
The convex non-empty subsets of {+, %, −} are: {+}, {−}, {%}, {+,%}, {%,−}, {+,%,−} — six options.
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A bias profile satisfies Convexity if all 6 cells are convex.
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{+, −} is ruled out by Convexity.
{+, %} and {%, −} are convex.
{+, %, −} is convex (the full set).
Convexity yields 6⁶ = 46656 profiles.
Narrow Epistemic Choice: epistemic bias is either {+^ep} or {+^ep, −^ep, %^ep} (the full set).
"the number of epistemic bias options is rather narrow, that is, we predominantly find {+^ep} or {+^ep,−^ep,%^ep}"
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A bias profile satisfies Narrow Epistemic Choice if all 3 epistemic cells use either {+} or {+,%,−}.
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Narrow Epistemic Choice alone yields (7 × 2)³ = 2744 profiles.
The 4 evidential options surviving Static Complementarity + Convexity: {+,%}, {%,−}, {%}, {−}.
These are the convex subsets minus {+}, {+,%,−} (which are the epistemic options) and minus {+,−} (non-convex).
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A bias profile satisfies Static Complementarity if:
- Epistemic cells use {+} or {+,%,−} (Narrow Epistemic Choice)
- Evidential cells use {+,%}, {%,−}, {%}, or {−}
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Static Complementarity + Convexity yields (4 × 2)³ = 512 profiles.
[1] English V1-Interrogative (Appendix A [1], from Sudo 2013:284).
PPQ: ⟨{+, %}, {+, −, %}⟩ IN-NPQ: ⟨{−}, {+}⟩ ON-NPQ: ⟨{−, %}, {+}⟩
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[2] Japanese ∅-Interrogative (Appendix A [2], from Sudo 2013:285).
PPQ: ⟨{%}, {+, −, %}⟩ IN-NPQ: ⟨{−}, {+, −, %}⟩ ON-NPQ: ⟨{+, %}, {+}⟩
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[3] Japanese no-Interrogative (Appendix A [3], = ex. (4), from Sudo 2013:288).
PPQ: ⟨{+}, {+, −, %}⟩ IN-NPQ: ⟨{−}, {+}⟩ ON-NPQ: ⟨{+, −, %}, {+}⟩
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[4] Japanese desho-Interrogative (Appendix A [4], = ex. (23), from Sudo 2013:290).
PPQ: ⟨{+, −, %}, {+}⟩ IN-NPQ: ⟨{+, −, %}, {−}⟩ ON-NPQ: ⟨{−, %}, {−}⟩
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[5] Hungarian ∧-Interrogative (Appendix A [5], from Gyuris 2017: Section 4).
PPQ: ⟨{+, %}, {+, −, %}⟩ IN-NPQ: ⟨{−}, {+}⟩ ON-NPQ: ⟨{−, %}, {+}⟩
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[6] Hungarian e-Interrogative (Appendix A [6], = ex. (10), from Gyuris 2017: Section 4). IN-NPQ is not expressible.
PPQ: ⟨{%}, {+, −, %}⟩ ON-NPQ: ⟨{%}, {+}⟩
- ppqEv : BiasChoice
- ppqEp : BiasChoice
- onNpqEv : BiasChoice
- onNpqEp : BiasChoice
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English V1 satisfies No Uniformity.
English V1 satisfies Narrow Epistemic Choice.
English V1 satisfies Static Complementarity.
Hungarian ∧-Interrogative has the same bias profile as English V1 (Appendix A: [5] = [1]).
Japanese ∅-Interrogative satisfies No Uniformity.
Japanese ∅-Interrogative violates PPQ ≠ NPQ: PPQ^ep = IN-NPQ^ep = {+,−,%}. Under the AND interpretation (both ev and ep must differ), identical epistemic values suffice to violate the constraint.
Japanese ∅-Interrogative satisfies Convexity.
Japanese ∅-Interrogative satisfies Static Complementarity: all ev cells ∈ {{+,%},{%,−},{%},{−}} and all ep cells ∈ {{+},{+,−,%}}. Despite violating PPQ ≠ NPQ, its profile is within the 512-profile SC-permissible space.
Japanese no-Interrogative satisfies No Uniformity.
Japanese no-Interrogative satisfies PPQ ≠ NPQ.
Japanese no-Interrogative violates Distributive Markedness: |PPQ^ev| = 1 < |ON-NPQ^ev| = 3. This is a known counterexample noted by §2.3.
Japanese no-Interrogative violates Static Complementarity: ON-NPQ^ev = {+,−,%} which is not in the static complementarity set of evidential options.
Japanese desho-Interrogative violates Avoid Disagreement: IN-NPQ^ev contains + and PPQ^ev contains −.
Japanese desho-Interrogative violates Narrow Epistemic Choice: IN-NPQ^ep and ON-NPQ^ep select {−}, which is neither {+} nor {+,%,−}.
Japanese desho-Interrogative violates PPQ ≠ NPQ: PPQ^ev = IN-NPQ^ev = {+,−,%}.
Japanese desho-Interrogative violates Static Complementarity (via narrowEpistemic failure).
English V1 violates Avoid Disagreement: PPQ^ep = {+,−,%} contains −, and IN-NPQ^ep = {+} contains +. This exemplifies the systematic incompatibility between Narrow Epistemic Choice and Polarity Match for epistemic cells (§3.1.2).
English V1 violates Don't Rule Out Agreement: IN-NPQ^ep = {+} does not contain −. Again, the epistemic dimension conflicts with NEC-derived empirical patterns (§3.1.2).
English V1 satisfies Distributive Markedness.
English V1 satisfies Collective Markedness.
Map G&G's evidential bias choice to Romero/BiasedPQ ContextualEvidence compatibility. A bias choice lists which evidence types are felicitous.
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Map G&G's epistemic bias choice to Romero/BiasedPQ OriginalBias compatibility.
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- Phenomena.Questions.Studies.GartnerGyuris2017.epistemicallyCompatible bc Semantics.Modality.BiasedPQ.OriginalBias.forP = List.contains bc Phenomena.Questions.Studies.GartnerGyuris2017.BiasValue.pos
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English V1 IN-NPQ (= Romero's LoNQ) requires evidence against p, matching Romero's Table 2 prediction.
English V1 ON-NPQ (= Romero's HiNQ) has epistemic bias for p only, matching Romero's requirement that HiNQ conveys original bias for p.
English V1 PPQ (= Romero's PosQ) is compatible with evidence for p or neutral evidence, matching Romero's Table 2.
Hungarian e-Interrogative has evidential "anti-bias" {%^ev} for PPQ: requiring neutral evidence only. This is the key counterexample to PPQ ≠ NPQ noted by §2.2.
This contrasts with standard PPQs which admit positive evidence.
Czech bias profile in G&G format, derived from @cite{simik-2024} Table 2
via czechBiasProfile.
Czech V1-Interrogative (InterPPQ/InterNPQ as PPQ/ON-NPQ):
- InterPPQ = PPQ: ev={%}, ep={+,%}
- DeclNPQ = IN-NPQ: ev={−}, ep={+,%}
- InterNPQ = ON-NPQ: ev={+,%,−}, ep={+,%}
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Czech PPQ (InterPPQ) admits only neutral evidence — narrower than English V1 PPQ. Czech InterPPQ is the default unbiased PQ, felicitous only when there is no compelling evidence either way.
Czech ON-NPQ (InterNPQ) has broader evidential distribution than English — it admits +, %, and − evidence, reflecting FALSUM^CZ's weaker requirements (@cite{simik-2024} §5).
Czech ON-NPQ (InterNPQ) epistemic bias admits + and % (speaker believes p or is neutral). Unlike English HiNQ which requires bias for p, Czech InterNPQ is also felicitous in explanation-seeking contexts (neutral epistemic bias, @cite{simik-2024} §5.2).
Czech V1 profile satisfies No Uniformity.
Czech V1 profile violates PPQ ≠ NPQ: PPQ^ep = IN-NPQ^ep = {+,%}. Czech InterPPQ and DeclNPQ share the same epistemic bias distribution, reflecting that both forms are felicitous when the speaker either believes p or is neutral.
Czech ON-NPQ evidential {+,%,−} violates Static Complementarity — this is expected because Czech FALSUM^CZ has broader distribution than English FALSUM, allowing all evidence types. The G&G framework was designed for English/Japanese/Hungarian where ON-NPQ evidence is narrower.
The key Czech vs English difference: Czech ON-NPQ admits positive evidence while English ON-NPQ does not. This is the empirical core of @cite{simik-2024}'s FALSUM^CZ proposal.