@cite{cummins-franke-2021}: Argumentative Strength of Numerical Quantity #
@cite{cummins-franke-2021}
Formalizes the conference registration scenario (C&F pp. 7–8) demonstrating that semantic and pragmatic argumentative strength can reverse the ordering of "more than M" expressions, and the REF 2014 case study on strategic use of "top M" claims (C&F §5, pp. 10–14).
Key Results #
- Semantic measure: "more than 110" > "more than 100" for goal "success" (because "more than 110" concentrates probability mass in goal-worlds)
- Pragmatic reversal: with 90% enrichment, the ordering flips — "more than 100" becomes pragmatically stronger (C&F Eq. 25)
- REF case study: universities use round "top M" claims and prefer the ranking measure on which they score better (C&F §5)
See also MacuchSilvaEtAl2024.lean for the experimental follow-up on
strategic quantifier choice in exam scenarios.
Methodology #
All ordinal comparisons use Bayes factor ratios (exact ℚ arithmetic), which are equivalent to comparing argStr values since log is monotone. This avoids dependence on log approximations.
"More than M" is equivalent to lower-bound meaning at M+1.
moreThan(M)(n) = true ↔ n > M ↔ n ≥ M+1 = lowerBound(M+1)(n)
These theorems ground the conference scenario's semantics in the
canonical moreThanMeaning from Numeral.Semantics. The conference
scenario (§2) uses probability counts directly for tractability, but
the underlying denotation is the same.
Conference argumentative goal
Equations
Instances For
P("more than 100" | G) = 1: all goal-worlds (value > 120) satisfy value > 100.
Instances For
P("more than 100" | ¬G) = 20/120 = 1/6: among ¬G worlds (value in [0, 120]), values > 100 span (100, 120], measure 20 out of 120. (C&F p. 7)
Instances For
P("more than 110" | G) = 1: all goal-worlds satisfy value > 110.
Instances For
P("more than 110" | ¬G) = 10/120 = 1/12: among ¬G worlds, values > 110 span (110, 120], measure 10 out of 120. (C&F p. 7)
Equations
Instances For
"More than 110" has a higher Bayes factor than "more than 100" for the conference goal.
BF("mt110") = 1/(1/12) = 12 > BF("mt100") = 1/(1/6) = 6.
This is C&F's key semantic result: the more precise (higher M) expression has higher argStr because it has lower P(u|¬G). (C&F p. 7)
Both utterances have positive argumentative strength for the goal.
P_A("more than 100" | G): 90% × P(value ∈ (100,150] | value ∈ (120,200]) + 10% × P(value > 100 | value ∈ (120,200]) = 90% × 30/80 + 10% × 1 = 27/80 + 8/80 = 35/80. (C&F p. 8)
Instances For
P_A("more than 100" | ¬G) = 1/6: Both enriched and literal paths give the same probability, since (100,150] ∩ [0,120] = (100,120] and literal >100 in [0,120] is also (100,120]. Total: 90% × 1/6 + 10% × 1/6 = 1/6. (C&F p. 8)
Instances For
P_A("more than 110" | G) = 1/10: 90% × P(value ∈ (110,120] | value ∈ (120,200]) + 10% × P(value > 110 | value ∈ (120,200]) = 90% × 0 + 10% × 1 = 1/10. The enriched range (110,120] doesn't intersect the goal range (120,200]. (C&F p. 8)
Instances For
P_A("more than 110" | ¬G) = 1/12: Both paths give P(value > 110 | value ∈ [0,120]) = 10/120 = 1/12. Total: 90% × 1/12 + 10% × 1/12 = 1/12. (C&F p. 8)
Instances For
PRAGMATIC REVERSAL: "more than 100" becomes pragmatically stronger.
BF_prag("mt100") = (35/80)/(1/6) = 21/8 = 2.625 BF_prag("mt110") = (1/10)/(1/12) = 6/5 = 1.2
This is C&F's central result (p. 8). Semantically mt110 > mt100, but pragmatically mt100 > mt110. The enrichment of "more than 110" to "not more than 120" makes it nearly unassertable in goal-worlds (P_A = 1/10), while "more than 100" enriched to "not more than 150" retains substantial assertability (P_A = 35/80).
Both utterances retain positive pragmatic argumentative strength.
Structural theorem: when P_a(u|¬G) increases (through wider enrichment), the Bayes factor decreases and thus argStr decreases. This is the mechanism behind the pragmatic reversal.
Claim type: absolute rank ("top M") or percentile ("top M per cent")
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
C&F examples 29–38: UK universities' "top M" claims from REF 2014 reports.
All claimed M values are round numbers (multiples of 5 or 10). Data verified against C&F p. 12. REF 2014 had 154 multi-subject institutions.
Equations
- One or more equations did not get rendered due to their size.
Instances For
H1 verification: all claimed M values are round (multiples of 5)
H1 verification: absolute claims are truthful (actual rank ≤ claimed M)
H1 verification: percentile claims are truthful. REF 2014 had 154 institutions; top 10% = rank ≤ 16, top 25% = rank ≤ 39.
H2 data: ranking measure preference (C&F p. 13).
Of 39 institutions with data, 19 ranked higher on GPA and 19 on power (Durham 20th on both). Institutions systematically prefer the measure on which they rank better.
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
GPA-preferred group: 9 cite GPA, 0 cite power, 10 cite neither. (p < 0.01, sign test; C&F p. 13)
Equations
- Phenomena.Persuasion.Studies.CumminsFranke2021.h2_gpaGroup = { groupSize := 19, citedPreferred := 9, citedNonPreferred := 0, citedNeither := 10 }
Instances For
Power-preferred group: 11 cite power, 2 cite GPA, 6 cite neither. (p < 0.05, sign test; C&F p. 13)
Equations
- Phenomena.Persuasion.Studies.CumminsFranke2021.h2_powerGroup = { groupSize := 19, citedPreferred := 11, citedNonPreferred := 2, citedNeither := 6 }
Instances For
H2: in each group, more institutions cite their preferred measure than the non-preferred one.
H2: group counts are internally consistent.