Intersective modifier

When a modifier is intersective, its contribution to the sentence's truth conditions do not depend on the particular expression it modifies. This means that one can test whether a modifier is intersective by seeing whether it gives rise to valid reasoning patterns such as the following.

1. Floyd is a Canadian surgeon.
2. Floyd is an arsonist.
3. Valid: Therefore Floyd is a Canadian arsonist.

With a non-intersective modifiers such as "skillful", the equivalent deduction would not be valid.

1. Floyd is a skillful surgeon.
2. Floyd is an arsonist.
3. Not valid: Therefore Floyd is a skillful arsonist.

Modifiers can be ambiguous, having both intersective and nonintersective interpretations. For instance, the example below has an intersective reading on which Oleg is both beautiful and a dancer, but it also has a merely subsective reading on which Oleg dances beautifully but need not himself be beautiful.

1. Oleg is a beautiful dancer.

On a textbook semantics for modification, an intersective modifier denotes the set of individuals which have the property in question. When the modifier modifies a modifiee which also denotes a set of individuals, the resulting phrase denotes the intersection of their denotations.

1. ${\displaystyle [[\text{blue}]]=\{x|x\text{ is blue }\}}$
2. ${\displaystyle [[\text{blue pig}]]=\{x|x\text{ is blue }\}\cap \{x|x\text{ is a pig }\}}$

Such meanings can be composed either by introducing an interpretation rule Predicate Modification which hard-codes intersectivity. However, this mode of composition can also be delivered by standard Function Application if the modifier is given a higher semantic type, either lexically or by applying a type shifter.

1. Predicate Modification Rule: If ${\displaystyle \alpha }$ is a branching node with daughters ${\displaystyle \beta }$ and ${\displaystyle \gamma }$ where ${\displaystyle [[\beta ]],[[\gamma ]]\in {\mathcal{D}}_{⟨e,t⟩}}$, then ${\displaystyle [[\alpha ]]=\lambda x_e.x\in [[\beta ]]\cap [[\gamma ]]}$.