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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.

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.