Formal semantics

Formal semantics describes linguistic meaning with explicit mathematical representations and rules for combining them. A truth condition states when a declarative expression is true in a model; compositional semantics explains how meanings depend on both the parts and their arrangement. A syntactic category alone need not determine the semantic operation.^[ref]

In a toy model, let $B=\{1,2\}$ denote the boxes and $R=\{2,3\}$ the red objects. “Every box is red” is true exactly when $B\subseteq R$, which fails here. “Some box is red” is true when $B\cap R\ne \varnothing$, which holds. The difference comes from the quantifier's rule, not a change in the noun's denotation. Such finite examples make competing semantic proposals testable while leaving conversational interpretation to additional assumptions about context and pragmatics.

Compositional semantics explains a complex expression's meaning through the meanings of its parts and the way syntax combines them. It requires a rule of combination, not merely a list of word meanings. This helps explain how a speaker can understand unfamiliar combinations of familiar material.^[ref]

In an original toy model, let the red objects be $R=\{1,2\}$ and round objects $O=\{2,3\}$. Treating red and round as an intersection gives $R\cap O=\{2\}$. Replacing and with or calls for a different combination, giving $R\cup O=\{1,2,3\}$. This stipulated analysis makes the connective's contribution explicit. Natural expressions require more care: not every adjective combines with a noun by intersection. A proposed formal semantics must specify its rules and check their predictions against interpretations.

A truth condition specifies what must hold for a declarative expression to be true under an interpretation and context. It is distinct from the expression's actual truth value: someone can understand what would make a statement true without knowing whether those circumstances obtain.^[ref]

For a toy example, interpret Every marked tile is blue using a set $M$ of marked tiles and a set $B$ of blue tiles. Its truth condition is $M\subseteq B$. If $M=\{1,2\}$ and $B=\{2,3\}$, it is false; changing $B$ to $\{1,2,3\}$ makes it true. The rule remains fixed while the model changes. Truth conditions allow formal semantics to separate an interpretation from the evidence used to assess a particular situation, although pragmatics can contribute further conversational meaning.