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An adjective is a part of speech characteristically used to express properties and modify nouns. It can also occur in predicative constructions, as in The cube is red. Its classification depends on its grammatical behavior, beyond whether its meaning sounds descriptive.^[ref]

In an original simple formal semantics, let red objects form $R=\{1,2\}$ and cubes form $C=\{2,3\}$. The expression red cube can denote the intersection $R\cap C=\{2\}$. This illustrates how an adjective can contribute a property to a noun phrase, while leaving the grammatical classification distinct from that particular semantic rule.

Modification alone is insufficient evidence for adjective status. In the Universal Dependencies scheme, a noun used to modify another noun in a compound remains tagged as a noun. Clear category criteria make linguistic annotation reproducible instead of treating every modifier as the same class.

Ai Taniguchi is a linguist specializing in formal semantics and pragmatics. Her University of Toronto Mississauga profile describes research on kinds of meaning beyond an utterance's literal main assertion, including social meaning and how context changes interpretation.^[ref]

This distinguishes what a sentence directly asserts from information conveyed through its use and context. Formal models make those proposed differences explicit enough to compare against linguistic examples. She is a coauthor of Essentials of Linguistics (2nd edition) by Catherine Anderson and others and uses comics and visual art for public communication about linguistics, as documented in the same institutional profile.

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.

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.

Grammatical aspect presents a temporal view of an event or state, for example as ongoing or as a whole. It concerns how the event is viewed, whereas grammatical tense concerns its relation to a temporal reference point. Affixes and auxiliary constructions can express aspectual distinctions.^[ref]

Compare “The train arrived” with “The train was arriving”. Both locate the situation in the past, but the second presents an unfolding situation and does not by itself assert a completed arrival. Comparing “is arriving” with “was arriving” instead changes tense while retaining the progressive construction. Such contrasts help isolate which part of an interpretation comes from each grammatical choice. A formal semantics can encode those differences explicitly, rather than treating every temporal expression as the same kind of label.

In a traditional subject-predicate analysis, a predicate is the portion of a clause that says something about its subject. It can include a verb together with other material, rather than being identical to a single verb form.^[ref]

For The sample changed color, the subject is the sample and the predicate is changed color. In The sample is red, is red expresses the predication, although the central property is contributed by the adjective red. The examples show why “predicate” and “action word” are not interchangeable classifications.

An explicit model can distinguish the grammatical expression from its interpretation. A formal semantics might represent the property red by a set and ask whether the sample belongs to it, while the syntax describes how is red combines with the subject. The linguistic unit and the mathematical representation are related analytical choices.

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.