@cite{cariani-santorio-wellwood-2024}: Confidence Reports #
@cite{cariani-santorio-wellwood-2024}
States-based semantics for nominal and adjectival confidence reports
(Ann is/has confident/confidence that p) and their comparative forms.
The paper extends Wellwood's @cite{wellwood-2015} cross-categorial
comparative analysis to gradable attitude expressions; the central
contribution is a POS-morpheme-free account of the positive form
(CSW §3.3) plus a per-holder, non-probabilistic confidence ordering
(CSW §4.1) that admits Tversky–Kahneman conjunction fallacies (CSW §4.6).
The substrate machinery lives in Theories/Semantics/Gradability/StatesBased.lean
(positive-region predicates over a preorder) and
Theories/Semantics/Attitudes/Confidence.lean (ConfidenceOrdering,
confidentEntry/certainEntry, the §4.6 logic theorems). This study
file connects CSW's empirical claims to the substrate theorems and
witnesses the central cross-framework disagreement against
Theories/Semantics/Attitudes/EpistemicThreshold.lean.
Coverage #
| CSW section | What this file covers |
|---|---|
| §3.3 | POS-free positive form: inPositiveRegion over confidentEntry |
| §4.6 (52) | Conjunction fallacy compatibility (Confidence.conjunction_fallacy_compatible) |
| §4.6 (53) | Upward monotonicity (Confidence.confidence_upward_monotone) |
| §4.6 (54) | Transitivity of comparative confidence |
| §4.6 (55) | Antisymmetry of equative confidence |
| §4.6 (56–58) | Connectedness — formalized as agnostic, per CSW p.27 |
| §4.6 (52) ↔ Threshold | Cross-framework refutation: EpistemicThreshold.confidence_not_probabilistic |
| §5.2 (65–66) | Asymmetric entailment certain ⊨ confident |
| §5.2 (63a–c) | Doubts triangle: confident + doubts mutually exclusive |
| §5.2 (72) | Comparative scale-mate equivalence |
| Wellwood 2015 → CSW | Compositional bridge under unique-state assumption |
Out of scope (future-work sections at the bottom) #
- §5.1 conditional confidence: CSW p.28 explicitly note this is "not entirely predicted by the system we have set up" and propose two off-the-shelf modifications without choosing between them.
- §5.3
likely: CSW sketch but do not formalize the extension to probability operators. The Moore-paradox asymmetry CSW discuss (74)–(75) is a synthesis claim across CSW + a separatelikelysemantics, not a CSW-derived prediction. - §3.5 varieties (
confident in Bill, bareconfident,feel confident): CSW's §4.5 distributional argument for the Neodavidsonian framework.
§1. Felicity Gradient #
CSW use a graded inventory of acceptability marks (✓ / ? / ?? /
#). Encoding judgments as a 4-valued enum preserves the gradient
that a Bool encoding flattens — the difference between ?? (CSW 65b
adjectival) and ? (CSW 66b nominal) is itself part of the data CSW
present.
Equations
- Phenomena.Gradability.Studies.CarianiSantorioWellwood2024.instDecidableEqFelicity x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Equations
- One or more equations did not get rendered due to their size.
Instances For
§2. Asymmetric Entailment: certain ⊨ confident (CSW (65)/(66)) #
CSW (65a) "Ann is confident that p, but she isn't certain that p." ✓ CSW (65b) "??Ann is certain that p, but she isn't confident that p." CSW (66a) "Bob has confidence, but not certainty, that p." ✓ CSW (66b) "?Bob has certainty, but not confidence, that p."
The adjectival pair (65) is more sharply contrasted (??) than the
nominal pair (66) (?). Both directions are encoded; the substrate
predicts the direction from certainEntry's maximality assumption.
The (65a)/(66a) felicitous pair: confidence without certainty is
consistent. Predicted by confident_not_entails_certain: when
confPt is strictly below the certainty contrast point, the
confidence positive region is not contained in the certainty one.
The (65b)/(66b) infelicitous pair: certainty without confidence is
inconsistent. Predicted by certain_entails_confident: when maxPt
is the top of the ordering, the certainty positive region is contained
in any confidence region with confPt ≤ maxPt.
Empirical record of the (65)/(66) felicity gradient.
Keeps adjectival (65b) and nominal (66b) markings distinct rather
than collapsing both to unacceptable.
- conf_without_certain : Felicity
(65a) ✓
- certain_without_conf_adjectival : Felicity
(65b) ?? — adjectival pair, sharper contrast
- certain_without_conf_nominal : Felicity
(66b) ? — nominal pair, weaker contrast
Instances For
§3. Logic of Confidence Reports (CSW §4.6) #
§3.1 Transitivity (CSW (54)/(57)) #
CSW (54): comparative confidence is transitive. CSW (57) is contradictory because asserting its third clause negates the consequent of this entailment.
§3.2 Antisymmetry (CSW (55)) #
CSW (55): "at least as confident of p as q" + "at least as confident of q as p" → "equally confident of p and q".
§3.3 Connectedness (CSW (56)/(58)) — Agnostic #
CSW p.27: "We remain agnostic about whether Connectedness actually holds for confident and confidence." (58) is a candidate counterexample where some propositions might simply not be comparable.
The substrate models this by using Preorder (which doesn't require
totality) rather than LinearOrder. There is no theorem to prove on
either side: the agnosticism is the substantive content.
CSW remain agnostic about Connectedness. Encoded as a flag rather than a theorem to make the agnosticism formally visible.
- csw_committed : Bool
Whether CSW commit to Connectedness for confidence orderings.
Instances For
§3.4 Conjunction Fallacy (CSW (52)) #
CSW (52): it is consistent for "John is not confident that Linda is a banker" and "John is confident that Linda is a feminist banker" to be true together. Confidence orderings are not constrained to respect logical conjunction (CSW's central argument against probability-functional accounts; @cite{tversky-kahneman-1983}).
Witness imported from Confidence.conjunction_fallacy_compatible.
§3.5 Upward Monotonicity (CSW (53)) #
CSW (53): "σ is confident that p" + "σ is more confident of q than of p" → "σ is confident that q". Direct consequence of preorder transitivity through the contrast point.
§3.6 Doubts Triangle (CSW (63a)–(63c)) #
CSW (63a)→(63b)→¬(63c): confident and doubts are mutually
exclusive, when the doubt contrast point lies strictly below the
confidence contrast point on the holder's confidence ordering.
The triangle: (63a) certain(p) entails (63b) confident(p) (via
certain_entails_confident); (63b) is inconsistent with (63c)
doubts(p) (this theorem); so (63a) is inconsistent with (63c).
The substrate models doubts as a negative-polarity entry on the
same ConfidenceOrdering as confident/certain: same entry shape
(StatesBasedEntry), but consumers test inLowerRegion rather than
inPositiveRegion.
§4. Comparative Scale-Mate Equivalence (CSW (72)) #
CSW (72): "A is more confident that p than that q" and "A is more certain that p than that q" are truth-conditionally equivalent.
CSW p.31 explanation: the comparative discards the contrast function
and uses only the shared background ordering. The substrate captures
this architecturally — statesComparativeSem takes no
StatesBasedEntry parameter, so the contrast point that distinguishes
confident from certain is invisible to the comparative. The
prediction holds by construction.
The comparative μ-measure ordering does not depend on which
entry's positive region is being asked about — it sees only the
measure function and the states.
This is the substrate-level witness that CSW (72)'s scale-mate
equivalence is structural, not provable. The function signature
omits any entry parameter; pluralizing across confidentEntry
and certainEntry is moot because the function never sees them.
§5. POS-Free Positive Form (CSW §3.3) #
The central architectural commitment of the paper. CSW (28b)/(40):
the positive form g-ness_C(s) holds iff s ≿ contrast(g-ness) —
no covert pos morpheme is invoked.
The substrate (StatesBased.inPositiveRegion) implements this
directly: entry.contrastPoint ≤ s over the background preorder.
Different lexical entries (confidentEntry, certainEntry) on the
same ConfidenceOrdering have different positive regions because
their contrastPoints differ — exactly CSW's analysis without ever
introducing POS.
POS-free positive form: confident and certain produce different
positive-region predicates on the same confidence ordering, with no
pos morpheme intervening.
The two predicates differ exactly when there is a state in
confident's region but not certain's — i.e., when the confidence
contrast point is strictly below the certainty contrast point.
§6. Cross-Framework Refutation: States-Based vs Threshold-Probabilistic #
CSW's central argument against extending threshold-style epistemic
semantics (Lassiter 2011/2016, Yalcin 2010) to confidence reports is
that confidence orderings need not respect logical conjunction
(CSW (52), §4.6). Probabilistic credence violates this: any monotone
credence function validates Pr(p ∧ q) ≤ Pr(p).
The two halves of the disagreement are now formal:
- States-based admits the fallacy:
Confidence.conjunction_fallacy_compatible(and the §3.4 invocation above). - Probabilistic credence forbids it:
EpistemicThreshold.prob_conjunction_elim. - A credence function realizing the fallacy cannot be probabilistic:
EpistemicThreshold.confidence_not_probabilistic.
This study file packages the disagreement as the joint statement below.
The empirical disagreement on CSW (52) / Tversky–Kahneman 1983, formalized as the conjunction of two opposing predictions:
- States-based prediction (CSW): there is a confidence ordering
admitting the fallacy (
conjunction_fallacy_compatibleprovides a Nat witness;confidence_not_probabilisticlifts it to anAgentCredencewitness). - Threshold-probabilistic prediction: any probabilistic credence
blocks the fallacy at every threshold
(
prob_conjunction_elim).
The two cannot agree on any datum where the fallacy is in fact consistent. CSW take the conjunction-fallacy data as decisive evidence against the threshold approach.
§7. Cross-Framework Agreement on certain #
Three independent treatments of certain agree that it sits at the
upper bound of an upper-bounded scale:
- Fragment (
Adjectives.certain.scaleType = .upperBounded) - Threshold (
EpistemicThreshold.EpistemicEntry.certain_.θ = 19/20, close to the scale max of 1) - States-based (
Confidence.certainEntry's contrast point is the ordering's maximum, byh_top)
The three encodings have genuinely different mathematical structure (enum tag vs ℚ value vs preorder maximality), so the agreement is not forced by a shared substrate primitive — it is a coincidence of independent commitments that nevertheless converges.
Two-way agreement: the Fragment and the threshold theory both
classify certain at the top of an upper-bounded scale.
Polarity asymmetry across the Fragment's confidence-scale entries:
confident/certain/sure pick out the upper region (positive
polarity, upperBounded); doubtful/unsure/uncertain pick out
the lower region (negative polarity, lowerBounded). The polarity
split lives in the scaleType field; the substrate's
inPositiveRegion vs inLowerRegion query then dispatches
accordingly.
§8. Compositional Bridge: Wellwood 2015 → CSW #
CSW build their analysis on Wellwood's @cite{wellwood-2015}
cross-categorial comparative. The bridge below specializes Wellwood's
adjectival_max_reduces (which proves the comparative reduces to a
direct degree comparison under unique-eventuality assumptions) to the
shape CSW use.
This is the only bridge in the file that does substantive composition — the others wire through substrate theorems directly.
Wellwood 2015's adjectivalComparative, instantiated for confidence
states, reduces to direct measure comparison under unique-state
assumptions. This closes CSW's compositionality claim that nominal
confidence and adjectival confident are interchangeable in
comparative form.
§9. Future Work #
Three CSW topics that this file does not formalize, with the reason each is deferred:
§5.1 Conditional Confidence (CSW (61)) #
CSW p.28: "Confidence reports interact with conditional antecedents in ways that are not entirely predicted by the system we have set up." CSW propose two off-the-shelf modifications (modal-base restriction or information-state indexing) and conclude (p.29): "Choosing between these options is, of course, beyond the scope of the present investigation." No theorem belongs here until CSW or successors choose between the two options.
§5.3 likely and the Moore-Paradox Asymmetry (CSW (74)–(75)) #
CSW sketch but do not formalize an extension to probabilistic modal
adjectives. The Moore-paradox asymmetry CSW illustrate is between
holder-relativized confident and impersonal likely. The
substrate's EpistemicThreshold.likely_ is agent-relative (cr a φ
threshold), not impersonal — so it cannot directly host the
asymmetry. A faithful formalization would require either (a) a
world-dependent objective probability primitive, or (b) a separate
study file anchored on Yalcin 2007 or Lassiter 2016 that introduces
the impersonal likely semantics.