# What is a conservative/intersective function?

I can't find any information on what a conservative or intersective function is. For any set A, a function F from P(A) into [P(A) -> {T,F}] is conservative iff....? I don't mean to ask for the raw answer, but any general ideas on where I can look would be greatly appreciated. Thank you!

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It sounds to me like the part following the iff is the definition. – JohnD Dec 13 '12 at 2:31
Well that part wasn't provided......if it was, I wouldn't need to be asking – pauliwago Dec 13 '12 at 2:37

## 1 Answer

From Edward L. Keenan, Some Logical Properties of Natural Language Quantifiers, in Joseph Almog and Paolo Leonardi, The Philosophy of David Kaplan:

Section 2.1.1, Def. 2 (with a slight change in notation):

$D$ from $\wp(E)$ into $[\wp(E)\to X]$, $X$ any set, is conservative iff for all $A,B,B'\subseteq E$, $D(A)(B)=D(A)(B')$ whenever $A\cap B=A\cap B'$.

Prop. 1, which they say is the usual definition:

A function $D$ from $\wp(E)$ to $[\wp(E)\to X]$ is conservative iff for all $A,B\subseteq E$, $DAB=DAA\cap B$.

(It appears that that last expression is short for $D(A)(B)=D(A)(A\cap B)$.)

Section 2.2.1, Def. 3:

$D$ is intersective iff for all $A,A',B,B'$, $D(A)(B)=D(A')D(B')$ if $A\cap B=A'\cap B'$.

$D$ here is a determiner, or generalized quantifier.

For more information, see the paper The Semantics of Determiners (pp. 11-13 for conservativity and intersectivity) and the brief survey Quantifiers: Semantics here. (Probably several other papers are relevant as well.)

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