View source for Intersective modifier

{{Short description|Utterance that conveys intersection of denotations}}

In [[linguistics]], an '''intersective modifier''' is an expression which [[grammatical modifier|modifies]] another by delivering the [[set intersection|intersection]] of their [[denotation]]s. One example is the English adjective "blue", whose intersectivity can be seen in the fact that being a "blue pig" [[entailment|entails]] 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.<ref>{{cite book |last=Morzycki|first= Marcin|date=2016|title=Modification |url=https://morzycki.github.io/work/papers/modification_book.pdf|publisher=Cambridge University Press |pages=14–16}}</ref><ref>{{cite encyclopedia |last=Kennedy |first=Chris |editor-last1=Graff Fara |editor-first1=Delia | editor-last2=Russell |editor-first2=Gillian |encyclopedia=The Routledge Companion to Philosophy of Language |title=Adjectives |url=http://semantics.uchicago.edu/kennedy/docs/routledge.pdf|year=2012 |publisher= Routledge}}</ref>

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 [[validity (logic)|valid]] reasoning patterns such as the following.<ref>{{cite book |last=Morzycki|first= Marcin|date=2016|title=Modification |url=https://morzycki.github.io/work/papers/modification_book.pdf|publisher=Cambridge University Press |page=16}}</ref>

# Floyd is a Canadian surgeon.
# Floyd is an arsonist.
# ''Valid:''  Therefore Floyd is a Canadian arsonist.

With a non-intersective modifiers such as "skillful", the equivalent [[deduction (logic)|deduction]] would not be valid.<ref>{{cite book |last=Morzycki|first= Marcin|date=2016|title=Modification |url=https://morzycki.github.io/work/papers/modification_book.pdf|publisher=Cambridge University Press |page=17}}</ref>

# Floyd is a skillful surgeon.
# Floyd is an arsonist.
# ''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 modifier|subsective]] reading on which Oleg dances beautifully but need not himself be beautiful.<ref>{{cite book |last=Morzycki|first= Marcin|date=2016|title=Modification |url=https://morzycki.github.io/work/papers/modification_book.pdf|publisher=Cambridge University Press |page=19}}</ref>

# Oleg is a beautiful dancer.

On a textbook semantics for modification, an intersective modifier [[denotation|denote]]s 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.<ref name = "hnk">{{cite book |last1=Heim |first1=Irene |last2=Kratzer |first2=Angelika |author-link1=Irene Heim |author-link2=Angelika Kratzer |date=1998 |title=Semantics in Generative Grammar|publisher=Blackwell |pages=63–68 |isbn=0631197133}}</ref>

# <math>[\![ \text{blue} ]\!] = \{x \, | \, x \text{ is blue }\}</math>
# <math>[\![ \text{blue pig} ]\!] = \{x \, | \, x \text{ is blue }\} \cap \{x \, | \, x \text{ is a pig }\}</math>

Such meanings can be [[compositionality|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 [[type theory|semantic type]], either lexically or by applying a [[type shifter]].<ref name = "hnk" />

# ''Predicate Modification Rule:'' If <math>\alpha</math> is a branching node with daughters <math>\beta</math> and <math>\gamma</math> where <math>[\![\beta]\!], [\![\gamma]\!] \in \mathcal{D}_{\langle e, t \rangle}</math>, then <math>[\![\alpha]\!] = \lambda x_e \, . \, x \in [\![\beta]\!]\cap[\![\gamma]\!]</math>.

== See also ==
{{wikt|intersective}}
* {{annotated link|Adjective}}
* {{annotated link|Grammatical modifier}}
* {{annotated link|Prepositional phrase}}
* {{annotated link|Relative clause}}
* {{annotated link|Subsective modifier}}
* {{annotated link|Privative adjective}}

== References ==
{{Reflist}}

{{lexical categories|state=collapsed}}

{{Formal semantics}}
[[Category:Grammar]]
[[Category:Semantics]]
[[Category:Adjectives by type]]
[[Category:Formal semantics (natural language)]]
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