Intensional Anaphora

The paper Intensional anaphora by Ezra Keshet and Steven Abney (2024) addresses the problem of anaphora to antecedents in intensional contexts, building on prior work by Stone (1999), Stone & Hardt (1999), and Brasoveanu (2010). The authors propose a new logic called Plural Intensional Presuppositional predicate calculus (PIP), which extends standard first-order predicate calculus with set abstraction and equality (PC) by adding a small number of defined constructs: unselective closure of bracketed variables, summation, formula-label definition and use, world subscripts on predicates, and presuppositions. PIP is designed to capture a broad range of complex anaphora phenomena, including donkey anaphora, quantificational subordination, modal subordination, paycheck pronouns, and summation pronouns, while also solving puzzles about intensional anaphora.

The paper begins by noting that intensional operators are often treated as quantifiers over possible worlds, parallel to determiners as quantifiers over individuals. However, individuals introduced in intensional contexts cannot serve as antecedents to later pronouns as easily as those introduced in quantificational contexts. For example, Everyone is eating a cheeseburger can be followed by They are large, but Andrea might be eating a cheeseburger does not support later anaphoric references like It is large or They are large. Stone (1999), Stone & Hardt (1999), and Brasoveanu (2010) solve this by restricting pronoun referents to exist in the world of evaluation. The authors show two types of counterexamples: (i) cases where anaphora is disallowed even when the pronoun's referents clearly exist, and (ii) cases where anaphora is allowed even though the pronoun's referents might not exist. They argue instead for a description-based account: A pronoun presupposes that its antecedent description has a non-empty extension (Section 1.3, hypothesis (9)). This presupposition must hold across the entire context set.

The paper presents several key examples. For instance, in a scenario with exactly five known candidates for mayor, There might already be a winner in the mayoral election cannot be followed by She is a woman or They are women, even though all candidates exist in the real world. Conversely, There must be some sort of animal in the shed. It's making quite a racket! is felicitous because the antecedent description animal in the shed has an extension in every world of the context set. The authors also discuss examples from corpora, such as If there's swelling, it's a Pancoast tumor that's metastasized and There must be a mistake, sir followed by And it's not the only one.

The PIP system is described in detail in Section 2. The domain consists of pluralities (sets of points), with worlds as singleton pluralities. Predicates take a world argument as a subscript. Modals are treated as relations between sets of possible worlds, with MIGHT and MUST defined in terms of existential and universal quantification over accessible worlds. Formula labels store antecedent descriptions for later use, and summation terms (Σxφ) produce pluralities by taking the union of individuals satisfying a formula. Pronouns are of two types: simple terms (variables) and summation terms. All pronouns presuppose that their denotations are non-empty, with singular pronouns further presupposing singularity. Felicity conditions are defined recursively, with presuppositions evaluated relative to the context set.

The analysis of intensional anaphora is presented in Section 3. The crux is that intensional anaphora cases involve a paycheck pronoun whose free variable is the world variable itself. For example, There might be a winner translates to MIGHTw(ΣwX) where X ≡ WINNERw([x]). When the pronoun she appears later, it denotes ΣxX, where the world variable w is bound at the discourse level, not inside the modal. This explains why the referent of a summation pronoun never comes from multiple worlds: the world subscripts inside the summation must share a single value. The felicity conditions require that the antecedent description have a non-empty extension in every world of the context set. For might sentences, there will be worlds where the description is empty, making the pronoun infelicitous. For must sentences, since the modal is realistic (includes the evaluation world), the description is guaranteed to be instantiated, making the pronoun felicitous.

The paper also addresses cases like Geach's Hob/Nob sentence (Hob thinks a witch has blighted Bob's mare, and Nob wonders whether she killed Cob's sow) and Partee's bathroom sentence (Either there's no bathroom in this house or it's in a funny place). For the former, PIP predicts a descriptivist analysis where the pronoun refers to the witch in Nob's belief worlds, with felicity requiring a unique witch in every accessible world. For the latter, the felicity conditions for disjunction ensure that the pronoun is felicitous only if the antecedent description is instantiated in all relevant worlds.

Section 4 reviews previous analyses. The authors critique Brasoveanu's IP-CDRT system as a value-based approach that requires pronoun referents to exist in the world of evaluation, which overgenerates (predicting felicitous anaphora in cases like the mayoral election) and undergenerates (predicting infelicity for must sentences like the animal in the shed). They also discuss recent hybrid approaches by Elliott (2022), Hofmann (2022), and Mandelkern (2022), which preserve more information about antecedent descriptions but do not address plurals, summation pronouns, or paycheck pronouns. The authors note that these systems are closer in spirit to PIP but require nontrivial extensions to match its coverage.

In conclusion, the paper argues that PIP, by combining formula labels, local variables, possible worlds, and presuppositions, provides a simple and general account of intensional anaphora that solves the puzzles identified by Stone (1999) and others. The key difference from previous approaches is the description-based nature of anaphora: pronouns refer to antecedent descriptions, not just values, and the presupposition that the description is instantiated in every world of the context set explains the observed felicity patterns. The system is grounded in standard predicate calculus with set abstraction, making it a classical foundation for this work.

Improvements to AI Systems Based on This Paper:

1. Enhanced Coreference Resolution in Intensional Contexts

• Improvement: Implement a presupposition-based anaphora resolver that checks whether an antecedent description has a non-empty extension across all worlds in the context set, rather than relying solely on existence in the evaluation world.

• What the improved system can do: Correctly reject anaphora like There might be a winner. She is a woman when the winner is not guaranteed in every accessible world, while accepting There must be an animal in the shed. It's making a racket because the description is instantiated in all relevant worlds. This reduces over- and under-generation in dialogue systems and story understanding.

2. Plural and Summation Pronoun Handling

• Improvement: Add support for summation terms (Σxφ) that union individuals across a formula, with felicity conditions requiring non-empty pluralities and singularity presuppositions for singular pronouns.

• What the improved system can do: Resolve paycheck pronouns (e.g., Every farmer who owns a donkey beats it → "it refers to the specific donkey per farmer) and summation pronouns (e.g., The students who passed got certificates. They were happy → they" refers to the union of all passing students) without conflating worlds, enabling more accurate multi-party dialogue and narrative comprehension.

3. Modal Subordination and World-Subscripted Predicates

• Improvement: Model modals as relations over sets of worlds, with predicates carrying explicit world subscripts, and bind world variables at the discourse level rather than inside the modal.

• What the improved system can do: Handle sequences like Andrea might be eating a cheeseburger. It would be large by tracking the world variable across sentences, correctly inferring that "it" refers to the cheeseburger in Andrea's belief worlds, not the actual world. This improves reasoning in hypothetical and counterfactual scenarios in question-answering and planning systems.

4. Presupposition Projection for Disjunction and Negation

• Improvement: Apply the recursive felicity conditions for complex formulas (e.g., disjunction, negation) to check that antecedent descriptions are instantiated in all worlds of the context set, not just the local evaluation world.

• What the improved system can do: Correctly judge sentences like Either there's no bathroom in this house or it's in a funny place as felicitous only when the bathroom's existence is guaranteed across all disjuncts, avoiding false presupposition failures in natural language inference tasks and dialogue state tracking.

5. Description-Based Anaphora for Intensional Antecedents

• Improvement: Replace value-based pronoun resolution (where the referent must exist in the evaluation world) with description-based resolution, storing antecedent descriptions via formula labels for later use.

• What the improved system can do: Resolve anaphora in cases where the referent does not exist in the actual world but does in all accessible worlds (e.g., There must be a mistake. It's not the only one) and reject cases where the referent exists but is not guaranteed (e.g., mayoral election example). This yields more robust coreference in legal, medical, and scientific text where hypothetical entities are discussed.

6. Unified Handling of Donkey Anaphora and Quantificational Subordination

• Improvement: Use unselective closure of bracketed variables and summation to unify donkey anaphora (e.g., Every farmer who owns a donkey beats it) with quantificational subordination (e.g., Most farmers own a donkey. They beat them.).

• What the improved system can do: Provide a single mechanism for resolving pronouns bound by quantifiers and those introduced in separate sentences, improving performance in reading comprehension and automated theorem proving where such constructions are common.

7. Handling of Intensional Paycheck Pronouns

• Improvement: Implement paycheck pronouns where the free variable is the world variable itself, allowing the pronoun to denote a description that varies across worlds (e.g., Hob thinks a witch blighted Bob's mare, and Nob wonders whether she killed Cob's sow).

• What the improved system can do: Correctly resolve cross-sentential anaphora in belief contexts, enabling AI systems to track entities across multiple agents' mental states, which is crucial for social reasoning, narrative generation, and multi-agent simulations.

8. Classical Logical Foundation for Compositional Semantics

• Improvement: Build the system on standard first-order predicate calculus with set abstraction, avoiding non-classical or dynamic logic frameworks.

• What the improved system can do: Integrate seamlessly with existing theorem provers and knowledge bases, allowing for verifiable reasoning about anaphora and modality in formal domains (e.g., legal reasoning, database querying with natural language interfaces) without requiring custom inference engines.

9. Felicity Condition Evaluation Across Context Sets

• Improvement: Define a recursive felicity function that evaluates presuppositions relative to the entire context set (set of accessible worlds), not just the current world.

• What the improved system can do: In dialogue systems, maintain a context set that updates with each utterance, enabling the system to predict whether a pronoun will be acceptable to a human listener, thus improving naturalness in conversational AI and reducing user confusion.

10. Handling of Must vs. Might Asymmetry

• Improvement: Encode that must is realistic (includes the evaluation world) while might is not, so that anaphora is always felicitous after must but only sometimes after might.

• What the improved system can do: Automatically generate or judge follow-up sentences in text generation, ensuring that AI-generated narratives maintain logical consistency when using modal verbs, e.g., generating There must be a solution. It is on the table but avoiding There might be a solution. It is on the table unless the solution is guaranteed.