The paper's symbol for a slot of a signature.
- symbol none = "◦"
- symbol (some Polarity.positive) = "+"
- symbol (some Polarity.negative) = "−"
Docs / Studies / Studies.MacCartneyManning2009
An extended model of natural logic
MacCartney, B. & Manning, C. D. (2009). In Proceedings of the Eighth International Conference on Computational Semantics (pp. 140–156)
MacCartney and Manning bring the implication signatures of Nairn, Condoravdi and Karttunen into natural logic through the relation that deleting an operator generates between a sentence and its complement, and a monotonicity class for each signature. Where the two slots of a signature do not imply the same polarity, the printed relation is the one whose constraints are the signature's implications, and inserting the operator generates its converse. The factive and the counterfactive are printed with the relation of their positive slot, since their full implications settle the complement; this is also why negating a factive sentence and its complement does not project the printed relation through negation, the negations being disjoint. Each signature's monotonicity class is that of the two-way implicative it weakens.
Errata
- The signature
−/◦is said to project^as|and both|and⌣as#, and is classed as downward monotone. An antitone operator that projects^as|projects⌣as|, so⌣cannot go to#.
TODO
- The projectivity table for the quantifiers and the inference procedure that chains edits.
- The English Fragment has no entries with the signatures of permit to and refuse to, and records the factivity of admit and pretend outside their implication signatures.
References
- MacCartney & Manning 2009, An extended model of natural logic
- Nairn et al. 2006, Computing Relative Polarity for Textual Inference
Reading the table
The slots of a signature.
- slots = [⊥, ↑Polarity.positive, ↑Polarity.negative]
The nine signatures under the paper's notation.
The definition is too large to print here; read it in the source.
Used by parse_signature_eq_implicative
The relations under the paper's symbols.
The definition is too large to print here; read it in the source.
The monotonicity classes, as the projectivity signatures +, − and •.
- classes = [("up", NaturalLogic.ProjectivitySignature.mono), ("down", NaturalLogic.ProjectivitySignature.anti), ("non", NaturalLogic.ProjectivitySignature.all)]
A row of the table: a signature, the relations that deleting and inserting an operator of it generate, and the signature's monotonicity class.
- signature : NaturalLogic.ImplicationSignature
- deletion : NaturalLogic.Relation
- insertion : NaturalLogic.Relation
- monotonicity : NaturalLogic.ProjectivitySignature
Used by Row.ofDatum, rows, deletion_eq, deletion_factive, monotonicity_eq
A row of the table from the paper's features.
The definition is too large to print here; read it in the source.
Built from
Used by rows, isSome_ofDatum
The table.
- rows = List.filterMap Row.ofDatum Examples.all
Built from ,
Used by deletion_eq, deletion_factive, negated_eq, monotonicity_eq
Built from
A negated example: the signature of its operator and the relation between the negated sentence and the negated complement.
- signature : NaturalLogic.ImplicationSignature
- relation : NaturalLogic.Relation
Used by NegatedRow.ofDatum, negatedRows, negated_eq, negated_factive
A negated example from the paper's features.
The definition is too large to print here; read it in the source.
Built from
Used by negatedRows, isSome_negatedRow_ofDatum
The negated examples.
- negatedRows = List.filterMap NegatedRow.ofDatum Examples.all
Built from ,
Used by negated_eq, negated_factive
Built from
Built from
Deletion and insertion
Where the two slots of a signature do not imply the same polarity, deleting an operator of the signature generates the relation whose constraints are the signature's implications, and inserting it generates the converse.
(r : Row) :
Built from ,
The factive and the counterfactive are printed with the relation of their implication in a positive context, their implications together constraining no relation.
(r : Row) :
Built from ,
Negation
Negating a sentence and its complement projects the relation that deleting the operator
generates through negation, an anti-morphism, exactly for the implicatives: he wasn't permitted to
live ⊏ he didn't live, but he didn't admit that he knew | he didn't know.
(n : NegatedRow) :
Built from , ,
{α : Type u_1} [BooleanAlgebra α] {x y : α} (h : ∀ a ∈ { positive := ↑Polarity.positive, negative := ↑Polarity.positive }.constraints, a.Holds x y) (n : NegatedRow) :
n ∈ negatedRows → n.signature = { positive := ↑Polarity.positive, negative := ↑Polarity.positive } → n.relation.Holds xᶜ yᶜ
Built from ,
Monotonicity
The monotonicity class of the two-way implicative a signature weakens: monotone below +/−,
whose operator is the identity, antitone below −/+, whose operator is negation, and none for
◦/◦ or a signature below neither.
The definition is too large to print here; read it in the source.
Used by monotonicity_eq
Each signature's monotonicity class is that of the two-way implicative it weakens.
Built from , ,
An antitone map between Boolean algebras that projects ^ as | projects ⌣ as | too.
{α : Type u_1} {β : Type u_2} [BooleanAlgebra α] [BooleanAlgebra β] {f : α → β} (hf : Antitone f) (h : ∀ (x y : α), NaturalLogic.Relation.negation.Holds x y → NaturalLogic.Relation.alternation.Holds (f x) (f y)) {x y : α} (hxy : NaturalLogic.Relation.cover.Holds x y) :
NaturalLogic.Relation.alternation.Holds (f x) (f y)