@cite{haslinger-2025-diss}: Imprecision and homogeneity

Empirical data from Nina Haslinger (2025), "Pragmatic constraints on imprecision and homogeneity," Doctoral Dissertation, Georg-August-Universität Göttingen (doi:10.53846/goediss-11395). The Sinn und Bedeutung 28 paper @cite{haslinger-2024} is the published version of dissertation chapters 3 and 5.

Two constraints

1. NO NEEDLESS MANNER VIOLATIONS (Ch 3-4): a tradeoff between structural complexity and semantic precision — more complex expressions must be more precise. The competition relation is potential p-equivalence (def (68), §3.3.3): two sentences compete iff there is some choice of issue parameter making their p-truth conditions equivalent.

2. INFERENCE PRESERVATION (Ch 6-7): an imprecise construal of φ must preserve the inferential relations between φ and its alternatives that hold on the precise construal.

Sub-namespaces

• Numerals (Ch 2, Ch 4) — round/non-round asymmetry, exactly/approximately modifiers, the German game-show paradigm.
• FormMeaning (Ch 3) — complexity-precision pairs: the/all-the, and/both, 100/exactly-100, blue/completely-blue.
• InferencePreservation (Ch 6, 7) — alternative-set blocking for non-round numerals, conjunctions, and numeral definites.

structure Phenomena.Imprecision.Studies.Haslinger2025.Numerals.NumeralImprecisionDatum : Type

Numeral imprecision datum: context-dependent exactness.

• numeral : Nat

  The numeral

• roundness : Core.Roundness.RoundnessGrade

  Roundness grade (from Core.Roundness)

• sentenceFrame : String

  Sentence frame

• exactContext : String

  Context favoring exact reading

• inexactContext : String

  Context favoring inexact reading

• actualValue : Nat

  Actual value in scenario

• acceptableExact : Bool

  Acceptable in exact context?

• acceptableInexact : Bool

  Acceptable in inexact context?

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprNumeralImprecisionDatum : Repr NumeralImprecisionDatum

Equations

• Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprNumeralImprecisionDatum = { reprPrec := Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprNumeralImprecisionDatum.repr }

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprNumeralImprecisionDatum.repr : NumeralImprecisionDatum → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.carsExact : NumeralImprecisionDatum

The cars scenario from @cite{haslinger-2025-diss} Ch 2.1.2 ex. (19) CARS(EXACT) and ex. (20) CARS(INEXACT), pp. 71-72.

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.carsNonRound : NumeralImprecisionDatum

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.Numerals.RoundnessAsymmetryDatum : Type

Minimal pair showing round/non-round asymmetry.

• roundNumeral : Nat

  Round numeral

• nonRoundNumeral : Nat

  Non-round numeral

• context : String

  Context (same for both)

• actualValue : Nat

  Actual value

• roundAcceptable : Bool

  Round numeral acceptable?

• nonRoundAcceptable : Bool

  Non-round acceptable?

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprRoundnessAsymmetryDatum.repr : RoundnessAsymmetryDatum → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprRoundnessAsymmetryDatum : Repr RoundnessAsymmetryDatum

Equations

• Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprRoundnessAsymmetryDatum = { reprPrec := Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprRoundnessAsymmetryDatum.repr }

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.hundredVsNinetyNine : RoundnessAsymmetryDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.fiftyVsFortyNine : RoundnessAsymmetryDatum

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.Numerals.GameShowDatum : Type

The game show scenario tests for homogeneity gaps.

Context makes both exact and inexact readings relevant. If numerals had gaps like plurals, neither sentence should be clearly true.

Source: @cite{haslinger-2025-diss} Ch 2.4.1 ex. (164), p. 72.

• sentence : String

  The sentence

• scenario : String

  Scenario description

• exactReadingTrue : Bool

  Exact reading true?

• inexactReadingTrue : Bool

  Inexact reading true?

• acceptable : Bool

  Judgment: is sentence acceptable?

• speakersAgree : Bool

  Do speakers agree?

• notes : String

  Notes

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprGameShowDatum.repr : GameShowDatum → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprGameShowDatum : Repr GameShowDatum

Equations

• Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprGameShowDatum = { reprPrec := Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprGameShowDatum.repr }

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.gameShowPositive : GameShowDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.gameShowNegative : GameShowDatum

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.Numerals.ExactlyModifierDatum : Type

"Exactly" removes imprecision, parallel to "all" for plurals.

Source: @cite{haslinger-2025-diss} Ch 4.2.1 ex. (4) and (9), pp. 174-176.

• bareSentence : String

  Bare numeral sentence

• exactlySentence : String

  Modified sentence

• context : String

  Context

• actualValue : Nat

  Actual value

• bareAcceptable : Bool

  Bare acceptable?

• exactlyAcceptable : Bool

  Exactly acceptable?

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprExactlyModifierDatum.repr : ExactlyModifierDatum → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprExactlyModifierDatum : Repr ExactlyModifierDatum

Equations

• Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprExactlyModifierDatum = { reprPrec := Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprExactlyModifierDatum.repr }

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.exactlyRemovesImprecision : ExactlyModifierDatum

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.Numerals.ApproximatelyDatum : Type

"Approximately" reduces imprecision by binding out the exact reading: bare 100 has both exact and inexact readings; approximately 100 has only inexact, so the set of available construals shrinks.

Source: @cite{haslinger-2025-diss} §4.2.1 ex. (4) and (10), p. 174-176; non-round case developed in §4.2.2 ex. (25)-(26).

• bareSentence : String

  Bare sentence

• approxSentence : String

  Approximately sentence

• roundness : Core.Roundness.RoundnessGrade

  Roundness grade (from Core.Roundness)

• approxNatural : Bool

  Is approximately acceptable with this numeral?

• notes : String

  Notes

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprApproximatelyDatum : Repr ApproximatelyDatum

Equations

• Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprApproximatelyDatum = { reprPrec := Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprApproximatelyDatum.repr }

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprApproximatelyDatum.repr : ApproximatelyDatum → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.approximatelyWithRound : ApproximatelyDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.approximatelyWithNonRound : ApproximatelyDatum

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.Numerals.Generalizations : Type

Core empirical generalizations about numeral imprecision.

• roundPermitsImprecision : Bool

  Round numerals permit imprecision

• nonRoundRequiresExact : Bool

  Non-round require exactness

• exactlyRemoves : Bool

  "Exactly" removes imprecision

• negationRequiresPolar : Bool

  Negation requires polar questions

• noHomogeneityGaps : Bool

  No clear homogeneity gaps (unlike plurals)

• contextSensitive : Bool

  Imprecision is context-sensitive

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprGeneralizations.repr : Generalizations → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprGeneralizations : Repr Generalizations

Equations

• Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprGeneralizations = { reprPrec := Phenomena.Imprecision.Studies.Haslinger2025.Numerals.instReprGeneralizations.repr }

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.generalizations : Generalizations

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.carsExamples : List NumeralImprecisionDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.roundnessAsymmetryExamples : List RoundnessAsymmetryDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.Numerals.gameShowExamples : List GameShowDatum

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.FormMeaning.ComplexityPrecisionPair : Type

A pair of expressions showing the complexity-precision tradeoff.

The potentiallyPEquivalent field encodes Haslinger's notion of potential p-equivalence (definition (68), §3.3.3): two sentences count as competitors for NO NEEDLESS MANNER VIOLATIONS iff there exists some choice of issue parameter making their p-truth conditions equivalent. This is weaker than raw truth-conditional equivalence and is not transitive (footnote on p. 88).

• lessComplexExpr : String

  Less complex expression

• moreComplexExpr : String

  More complex expression

• complexitySource : String

  What makes it more complex

• lessComplexImprecise : Bool

  Does less complex permit imprecision?

• moreComplexImprecise : Bool

  Does more complex permit imprecision?

• potentiallyPEquivalent : Bool

  Potential p-equivalence (Haslinger def 68): some issue-parameter choice makes the two p-truth-conditionally equivalent. NOT TC-equivalence.

• constructionType : String

  Construction type

def Phenomena.Imprecision.Studies.Haslinger2025.FormMeaning.instReprComplexityPrecisionPair.repr : ComplexityPrecisionPair → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.FormMeaning.instReprComplexityPrecisionPair : Repr ComplexityPrecisionPair

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.FormMeaning.doorsAllDoors : ComplexityPrecisionPair

Plural definites: "the doors" vs "all the doors". The German parallel (die Türen / alle Türen) holds, and §3.2.2 (Table 3.1) surveys 12+ language families showing the absence of the inverse pattern.

Source: @cite{haslinger-2025-diss} Ch 3.1.1 ex. (3), p. 86; gap-removal contrast ex. (4).

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.FormMeaning.andBoth : ComplexityPrecisionPair

Conjunctions: "Ann and Bert" vs "both Ann and Bert"

Source: @cite{haslinger-2025-diss} Ch 3.3 ex. (48), p. 104; further developed in §3.5.1 (p. 126).

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.FormMeaning.numeralExactly : ComplexityPrecisionPair

Numerals: "100 cars" vs "exactly 100 cars"

Source: @cite{haslinger-2025-diss} Ch 4.2 ex. (4), p. 174 + ex. (9), p. 176 (CARS(EXACT) scenario).

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.FormMeaning.blueCompletely : ComplexityPrecisionPair

Summative predicates: "blue" vs "completely blue"

Source: @cite{haslinger-2025-diss} on summative predicates.

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.FormMeaning.attestedPairs : List ComplexityPrecisionPair

Equations

• One or more equations did not get rendered due to their size.

inductive Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.InferenceRelation : Type

Inference relation between expression and alternative.

• entails : InferenceRelation
• contradicts : InferenceRelation
• independent : InferenceRelation

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprInferenceRelation.repr : InferenceRelation → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprInferenceRelation : Repr InferenceRelation

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instDecidableEqInferenceRelation : DecidableEq InferenceRelation

Equations

• Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instDecidableEqInferenceRelation x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

structure Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.NumeralBlockingDatum : Type

Numeral alternative blocking datum.

The alternatives of a numeral n include nearby numerals. If n contradicts m on precise reading, imprecise construal can't be compatible with m.

Source: @cite{haslinger-2025-diss} Chapter 6, @cite{sauerland-stateva-2007}

• numeral : Nat

  The numeral

• round : Bool

  Is it round?

• alternative : Nat

  Key alternative

• alternativeInSet : Bool

  Is alternative in the alternative set?

• inferenceRelation : InferenceRelation

  Inference relation (precise reading)

• imprecisionBlocked : Bool

  Is imprecision blocked?

• explanation : String

  Explanation

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprNumeralBlockingDatum.repr : NumeralBlockingDatum → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprNumeralBlockingDatum : Repr NumeralBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.ninetyNineBlocked : NumeralBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.hundredNotBlocked : NumeralBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.AlternativeSetAsymmetry : Type

The asymmetry depends on conventionalized alternative sets.

Round numbers have "coarse" alternative sets. Non-round numbers have "fine" alternative sets that include round neighbors.

Source: @cite{haslinger-2025-diss} Chapter 6

• roundNumeral : Nat

  Round numeral

• roundAlternatives : List Nat

  Its alternative set (simplified)

• nonRoundNumeral : Nat

  Non-round numeral

• nonRoundAlternatives : List Nat

  Its alternative set (simplified)

• explanation : String

  Asymmetry explanation

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprAlternativeSetAsymmetry.repr : AlternativeSetAsymmetry → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprAlternativeSetAsymmetry : Repr AlternativeSetAsymmetry

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.hundredNinetyNineAsymmetry : AlternativeSetAsymmetry

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.ConjunctionBlockingDatum : Type

Conjunction blocking datum.

Conjunctions have conjuncts as alternatives. "A and B are P" entails "A is P" and "B is P". Non-maximal reading would fail to preserve these entailments.

Source: @cite{haslinger-2025-diss} Chapter 7

• sentence : String

  The conjunction sentence

• conjuncts : List String

  The conjuncts

• alternatives : List String

  Alternatives (the conjuncts as sentences)

• entailmentHolds : Bool

  Entailment holds on precise reading?

• nonMaxPreservesEntailment : Bool

  Would non-max preserve entailment?

• nonMaxBlocked : Bool

  Is non-max blocked?

• explanation : String

  Explanation

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprConjunctionBlockingDatum.repr : ConjunctionBlockingDatum → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprConjunctionBlockingDatum : Repr ConjunctionBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.annBertConjunction : ConjunctionBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.threeStudentsConjunction : ConjunctionBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.NumeralDefiniteBlockingDatum : Type

Numeral-modified definites: "the four doors"

These have homogeneity gaps but resist non-maximality. The structural alternative is the numeral indefinite (not sub-numerals).

Source: @cite{haslinger-2025-diss} §7.2.3, eq. (11)-(16), p. 305; ftn. 7 p. 309 on the numeral indefinite's lack of homogeneity gap.

• sentence : String

  The sentence

• numeral : Nat

  The numeral

• indefiniteAlternative : String

  The structural alternative (numeral indefinite, no definite article)

• entailsIndefiniteAlternative : Bool

  Does precise definite reading entail the indefinite alternative?

• nonMaxEntailsAlternative : Bool

  Would non-max definite reading entail the indefinite alternative?

• nonMaxBlocked : Bool

  Is non-max blocked?

• explanation : String

  Explanation

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprNumeralDefiniteBlockingDatum.repr : NumeralDefiniteBlockingDatum → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprNumeralDefiniteBlockingDatum : Repr NumeralDefiniteBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.fourDoorsBlocking : NumeralDefiniteBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

structure Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.CollectiveCumulativeException : Type

Collective and cumulative predicates sometimes permit non-maximality even with conjunctions.

This is an exception that inference preservation needs to handle.

Source: @cite{haslinger-2025-diss} Chapter 7

• sentence : String

  The sentence

• predicateType : String

  Predicate type

• nonMaxPossible : Bool

  Is non-max possible?

• explanation : String

  Why?

@[implicit_reducible]

instance Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprCollectiveCumulativeException : Repr CollectiveCumulativeException

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.instReprCollectiveCumulativeException.repr : CollectiveCumulativeException → Nat → Std.Format

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.collectiveMeetException : CollectiveCumulativeException

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.cumulativeCarryException : CollectiveCumulativeException

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.numeralBlockingExamples : List NumeralBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.conjunctionBlockingExamples : List ConjunctionBlockingDatum

Equations

• One or more equations did not get rendered due to their size.

def Phenomena.Imprecision.Studies.Haslinger2025.InferencePreservation.collectiveExceptions : List CollectiveCumulativeException

Equations

• One or more equations did not get rendered due to their size.