6.2 Compositional semantics and sentence meaning

Principles of Compositional Semantics

Compositional semantics explains how we build sentence meaning from smaller parts. The core idea is straightforward: you know what individual words mean, and you know how grammar combines them, so you can figure out the meaning of any sentence you've never heard before. This is why you can understand a completely novel sentence like "The purple giraffe danced on a cloud" even though no one has ever said that to you.

This concept, called the principle of compositionality, was developed by the philosopher Gottlob Frege. It's foundational to modern semantics because it explains how a finite vocabulary can generate an infinite number of meaningful sentences.

Three ingredients make compositionality work:

• Lexical semantics — the meanings of individual words
• Syntactic structure — the grammatical rules for how words combine
• Semantic rules — the principles that derive meaning from those combinations

Application of Semantic Rules

Semantic rules are the specific operations that combine meanings as you move through a sentence's structure. The main ones to know:

• Function application — the most common rule. A predicate (like a verb) takes an argument (like a noun phrase) and produces a new meaning. For example, "loves" takes "John" and "Mary" to produce $loves(John, Mary)$.
• Predicate modification — combines two descriptive expressions that apply to the same thing. "Red" and "ball" both describe the same object, so they combine to give you red ball.
• Lambda abstraction — creates a function from an expression. The notation $\lambda x. loves(x, Mary)$ represents "the property of loving Mary," which is a function waiting for someone to fill in the $x$ slot.

The process for analyzing a sentence works bottom-up in four steps:

1. Draw out the syntactic structure (a tree diagram of the sentence)
2. Assign a semantic type to each word at the bottom of the tree ($e$ for entities like "John," $t$ for truth values, and $\langle e, t \rangle$ for functions from entities to truth values, like "runs")
3. Apply semantic rules at each node, combining meanings as you move upward
4. Arrive at the meaning of the full sentence at the top of the tree

So for "John runs," you'd take $John$ (type $e$) and $runs$ (type $\langle e, t \rangle$), apply function application, and get $runs(John)$ (type $t$), which is either true or false.

The formal notation uses lambda calculus and type theory. A two-place predicate like "loves" is written $\lambda x. \lambda y. loves(x, y)$, meaning it's a function that takes two arguments one at a time.

Syntactic Structure and Ambiguity in Semantics

Syntactic structure and sentence meaning

The syntax-semantics interface is where grammatical structure meets meaning. The way a sentence is structured determines which interpretations are possible, because structure maps grammatical roles (subject, object) onto thematic roles (agent, patient, experiencer, theme).

For example, in "The dog chased the cat," syntax tells you "the dog" is the subject, and the semantics maps that to the agent (the one doing the chasing). "The cat" is the object, mapped to the patient (the one being chased).

Meaning builds up hierarchically: words combine into phrases, phrases combine into clauses, and clauses combine into full sentences. At each level, semantic rules apply.

Structural ambiguity occurs when a single string of words can be parsed into more than one syntactic tree, producing different meanings. Take "I saw the man with binoculars." Two possible structures:

• I used binoculars to see the man — "with binoculars" modifies the verb "saw"
• I saw a man who had binoculars — "with binoculars" modifies the noun "the man"

Same words, different trees, different meanings. This is why syntax matters for semantics.

Resolving Ambiguity Through Semantics

Ambiguity shows up in several forms:

• Lexical ambiguity — a single word has multiple meanings. "Bank" could mean a financial institution or the edge of a river. Context usually resolves this quickly.
• Structural ambiguity — different syntactic parses yield different meanings, as above. "The chicken is ready to eat" could mean the chicken is prepared as food, or the chicken is hungry and about to eat.
• Scope ambiguity — arises when quantifiers can be interpreted in different orders. "Every child ate a cookie" could mean each child ate one (possibly different) cookie, or there's one specific cookie that every child ate. The two readings correspond to different scope orderings of "every" and "a."

Quantifier scope interactions are a major topic in formal semantics. "A teacher praised every student" is ambiguous between one teacher praised all the students and for each student, some teacher praised them.

Other sources of ambiguity include anaphora resolution, which is figuring out what a pronoun refers to. In "John told Bill he was smart," does "he" refer to John or Bill?

Techniques for resolving ambiguity include:

• Contextual clues from the surrounding discourse
• Pragmatic principles like Grice's maxims (e.g., speakers aim to be informative and relevant)
• Formal semantic analysis using tools like truth conditions — specifying exactly what would have to be true in the world for a sentence to be true. The classic example: "The king of France is bald" raises the question of whether this sentence is false or simply lacks a truth value, since France has no king.