🔠Intro to Semantics and Pragmatics Unit 12 – Formal Semantics: Montague & Dynamic

Key Concepts and Terminology

• Formal semantics studies the meaning of natural language expressions using mathematical and logical tools
• Montague grammar named after Richard Montague who developed a formal semantic theory for natural language
• Intensional logic extends first-order logic to include modal operators and lambda abstraction
• Type-driven interpretation assigns semantic types to syntactic categories and computes meanings compositionally
• Dynamic semantics focuses on the context-changing potential of utterances and how meaning evolves in discourse
• Discourse representation theory (DRT) represents the meaning of a discourse as a structured set of formulas
• Anaphora resolution determines the referents of pronouns and other anaphoric expressions within a discourse
• Presupposition refers to the implicit assumptions or background information required for an utterance to be meaningful

Historical Context and Development

• Formal semantics emerged in the late 1960s and early 1970s as a response to the limitations of earlier approaches to meaning
• Montague's work in the 1970s laid the foundation for modern formal semantics by providing a rigorous mathematical treatment of natural language
• Montague's "Universal Grammar" paper (1970) introduced the idea of a systematic correspondence between syntactic and semantic rules
• Dynamic semantics developed in the 1980s as an extension of Montague grammar to account for discourse-level phenomena
• Discourse Representation Theory (DRT) proposed by Hans Kamp in 1981 as a framework for representing the meaning of a discourse
• Irene Heim's File Change Semantics (1982) and Kamp's DRT were early influential dynamic semantic theories
• Subsequent work in dynamic semantics has refined and extended these initial approaches to cover a wider range of linguistic phenomena

Montague Grammar Basics

• Montague grammar is a formal semantic theory that assigns meanings to natural language expressions using a type-driven interpretation
• The basic principle of compositionality states that the meaning of a complex expression is a function of the meanings of its parts and the way they are combined
• Montague's system includes a syntactic analysis (based on categorial grammar) and a semantic interpretation (based on intensional logic)
• Syntactic categories (e.g., S, NP, VP) are mapped to semantic types (e.g., t, e, $\langle e,t \rangle$) via a homomorphism
• Lambda calculus is used to represent the meanings of functional expressions and enable compositional interpretation
  • Example: the semantic representation of "walks" might be $\lambda x. \text{walk}(x)$
• Intensional logic allows for the representation of modal concepts like necessity and possibility
  • Example: "John believes that Mary is happy" can be represented as $\text{believe}(j, \wedge \text{happy}(m))$

Formal Semantic Representations

• Formal semantics uses logical formulas to represent the meaning of natural language expressions
• First-order logic is used to represent the meaning of simple declarative sentences
  • Example: "John loves Mary" can be represented as $\text{love}(j, m)$
• Lambda calculus is used to represent the meaning of functional expressions like verbs and adjectives
  • Example: "red" can be represented as $\lambda x. \text{red}(x)$
• Generalized quantifiers are used to represent the meaning of determiners and quantificational noun phrases
  • Example: "every" can be represented as $\lambda P. \lambda Q. \forall x (P(x) \to Q(x))$
• Intensional logic is used to represent modal concepts and propositional attitudes
  • Example: "necessarily" can be represented as $\Box$
• Tense and aspect are often represented using temporal logic or event semantics
• Dynamic semantics extends these representations to capture the context-changing potential of utterances

Dynamic Semantics: Core Principles

• Dynamic semantics views meaning as context change potential rather than just truth conditions
• The meaning of an utterance is its ability to change the context, which includes the discourse referents and the information about them
• Discourse referents are entities introduced into the context by noun phrases and referred to by anaphoric expressions
• Anaphoric expressions like pronouns are interpreted relative to the current context and can access previously introduced discourse referents
  • Example: in "A man walks in the park. He whistles.", "he" refers back to the discourse referent introduced by "a man"
• The context is updated incrementally as the discourse unfolds, with each new utterance potentially modifying the context
• Dynamic semantics can account for various discourse phenomena like anaphora, presupposition, and donkey sentences
  • Example: in "Every farmer who owns a donkey beats it", "it" can refer back to the donkey introduced in the relative clause
• Different dynamic semantic theories (e.g., DRT, File Change Semantics) provide different formal mechanisms for representing and updating contexts

Compositional Analysis Techniques

• Compositional analysis in Montague grammar involves computing the meaning of a complex expression from the meanings of its parts
• The syntactic structure of an expression guides the semantic composition process
• Function application is the basic composition rule, where a functional expression is applied to its argument
  • Example: in "John walks", the meaning of "walks" ($\lambda x. \text{walk}(x)$) is applied to the meaning of "John" ($j$) to yield $\text{walk}(j)$
• Lambda abstraction is used to create functional expressions by abstracting over a variable
  • Example: from "walks" ($\text{walk}(x)$), we can abstract over $x$ to create $\lambda x. \text{walk}(x)$
• Quantifier raising is a technique used to handle quantificational noun phrases by raising them to a higher position in the logical form
  • Example: "every man walks" is analyzed as $\forall x (\text{man}(x) \to \text{walk}(x))$, with "every man" raised to take scope over the entire formula
• Type-shifting rules are sometimes employed to resolve type mismatches between expressions
• In dynamic semantics, composition rules are extended to update the context in addition to computing truth conditions

Applications in Natural Language Processing

• Formal semantic representations can be used in various natural language processing tasks
• Natural language inference (recognizing entailment and contradiction) can be performed by comparing the logical forms of sentences
  • Example: "John loves Mary" entails "John likes Mary" if we have the background knowledge that $\forall x \forall y (\text{love}(x,y) \to \text{like}(x,y))$
• Question answering systems can use formal semantics to derive the logical form of a question and match it against a knowledge base
• Dialogue systems can use dynamic semantics to keep track of the discourse context and resolve anaphoric references across utterances
• Machine translation can benefit from formal semantics by ensuring that the meaning of the source sentence is preserved in the target language
• Formal semantics can help in handling ambiguity and underspecification in natural language
• Combining formal semantics with machine learning techniques (e.g., semantic parsing) can improve the performance of NLP systems

Challenges and Limitations

• Fully capturing the meaning of natural language expressions in formal logic is challenging due to the complexity and flexibility of language
• Some aspects of meaning (e.g., vagueness, metaphor, irony) are difficult to represent in a formal system
• The principle of compositionality, while useful, may not always hold in natural language (e.g., idioms, conventionalized expressions)
• Acquiring large-scale lexical semantic resources (e.g., type hierarchies, selectional preferences) for use in formal semantic analysis is time-consuming and labor-intensive
• Integrating world knowledge and commonsense reasoning into formal semantic systems remains an open challenge
• Handling contextual factors like speaker intention, presupposition accommodation, and implicature calculation often requires going beyond the literal meaning of expressions
• The computational complexity of logical inference can be a bottleneck in practical applications of formal semantics
• There is a trade-off between the expressivity of the formal language used and its computational tractability