Intersective modifier

In <a href="/facts/Linguistics/oQuamvf2">linguistics</a>, an intersective modifier is an expression which <a href="/facts/Grammatical_modifier/BRk4Hoxs">modifies</a> another by delivering the <a href="/facts/Set_intersection/RPfUNJh9">intersection</a> of their <a href="/facts/Denotation/oABEtLtA">denotations</a>. One example is the English adjective "blue", whose intersectivity can be seen in the fact that being a "blue pig" <a href="/facts/Entailment/AyDXp4dn">entails</a> being both blue and a pig. By contrast, the English adjective "former" is non-intersective since a "former president" is neither former nor a president.
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 <a href="/facts/Validity_(logic)/oMGkywGC">valid</a> reasoning patterns such as the following.

<ol><li>Floyd is a Canadian surgeon.</li>
<li>Floyd is an arsonist.</li>
<li>Valid: Therefore Floyd is a Canadian arsonist.</li></ol>
With a non-intersective modifiers such as "skillful", the equivalent <a href="/facts/Deduction_(logic)/HadA1zWS">deduction</a> would not be valid.

<ol><li>Floyd is a skillful surgeon.</li>
<li>Floyd is an arsonist.</li>
<li>Not valid: Therefore Floyd is a skillful arsonist.</li></ol>
Modifiers can be <a href="/facts/Ambiguous/mgn4oEoQ">ambiguous</a>, 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 <a href="/facts/Subsective_modifier/L6l2NJlG">subsective</a> reading on which Oleg dances beautifully but need not himself be beautiful.

<ol><li>Oleg is a beautiful dancer.</li></ol>
On a textbook semantics for modification, an intersective modifier <a href="/facts/Denotation/oABEtLtA">denotes</a> 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.

<ol><li>
 
 
 
 [
 
 [
 
 blue
 
 ]
 
 ]
 =
 {
 x
 
 
 |
 
 
 x
 
  is blue 
 
 }
 
 
 {\displaystyle [\![{\text{blue}}]\!]=\{x\,|\,x{\text{ is blue }}\}}
 
</li>
<li>
 
 
 
 [
 
 [
 
 blue pig
 
 ]
 
 ]
 =
 {
 x
 
 
 |
 
 
 x
 
  is blue 
 
 }
 ∩
 {
 x
 
 
 |
 
 
 x
 
  is a pig 
 
 }
 
 
 {\displaystyle [\![{\text{blue pig}}]\!]=\{x\,|\,x{\text{ is blue }}\}\cap \{x\,|\,x{\text{ is a pig }}\}}
 
</li></ol>
Such meanings can be <a href="/facts/Compositionality/yP2hR4Le">composed</a> 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 <a href="/facts/Type_theory/xj0xzVqI">semantic type</a>, either lexically or by applying a <a href="/facts/Type_shifter/Jm1cDtFr">type shifter</a>.

<ol><li>Predicate Modification Rule: If 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 is a branching node with daughters 
 
 
 
 β
 
 
 {\displaystyle \beta }
 
 and 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 where 
 
 
 
 [
 
 [
 β
 ]
 
 ]
 ,
 [
 
 [
 γ
 ]
 
 ]
 ∈
 
 
 
 D
 
 
 
 ⟨
 e
 ,
 t
 ⟩
 
 
 
 
 {\displaystyle [\![\beta ]\!],[\![\gamma ]\!]\in {\mathcal {D}}_{\langle e,t\rangle }}
 
, then 
 
 
 
 [
 
 [
 α
 ]
 
 ]
 =
 λ
 
 x
 
 e
 
 
 
 .
 
 x
 ∈
 [
 
 [
 β
 ]
 
 ]
 ∩
 [
 
 [
 γ
 ]
 
 ]
 
 
 {\displaystyle [\![\alpha ]\!]=\lambda x_{e}\,.\,x\in [\![\beta ]\!]\cap [\![\gamma ]\!]}
 
.</li></ol>

Intersective modifier open-in-new

Intersective modifier