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\title{Function-induced information}
\author{R. Zuber\\ CNRS Paris\\ 2 place Jussieu,\\75005 Paris,
France \\ rz@ccr.jussieu.fr}
\date{}

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{\Large
\begin{center}
	\textbf{Function-induced information}
\end{center}
}

\section{Introduction}

It is known, in particular from the study
of generalized quantifiers, that in general,
not all functions with a given domain and range are necessary to
represent semantic information. Thus concerning for instance quantifiers
of the type $<1,1>$ there are various non-logical
constraints (conservativity, monotony, etc.) which can severly limit the
number of functions which are
used to convey or to represent semantic information. The purpose of
this paper is to show
that a particular semantic information can be associated with these
constraints on functions. It will also be shown that this information
corresponds in fact to some traditional type of information such as
\emph{presupposed information} and \emph{asserted information}. I will
first discuss propositional presuppositions and assertions of simple noun
phrases or rather of quantifiers of the type $<1>$. Then, using the notion of an
intersective function, I will generalize the
notion of presupposed and asserted information to any grammatical
category (or rather to any logical type). The basic idea expressed
informally is that presupposed
information of an expression interpreted by a function is independent of
the argument of that function and asserted information in some sense
depends only on the argument and less so on the function itself. To take
a classical example such as \emph{the king of France}, one notices that
the existence of the object denoted by this noun phrase is
(almost) independent of
the property it has. In other words the sentence of the form \emph{The
king of France VP} presupposes the existence of the king of France
independently of the VP. In this sense one can say that it is the NP
\emph{the king of France} which presupposes the existence of the king and
not the sentence in which this NP occurs. Similar considerations are valid
for assertions.


Although it will be made more precise in the third section, all the functions
we will be interested in are functions from Boolean algebras into Boolean
algebras, as we consider, following the work of Keenan and his
associates, that logical types (denotations of natural language
expressions) form Boolean algebras. In our considerations we will mainly
operate on the expressions of some simple logical language in which
symbols for generalized quantifiers occur. In particular sentences
of the form \textbf{Q(P)} or \textbf{D(S)(P)}, where \textbf{Q} is a
quantifier of the
type $<1>$, \textbf{S} and \textbf{P} are properties, and
\textbf{D} is a determiner or a
quantifier  of the type $<1,1>$, will often be used.

\section{Quantifier induced semantic information}

\subsection{Quantifier induced presupposition}

In this sub-section I will introduce the notion
of presupposition induced by a quantifier of the type $<1>$ by
considering a presupposition of the sentence of the form \textbf{Q(P)}. We have
the following definition:

(D1) Sentence \textbf{Q(P)} QI-presupposes (or has as a QI-(quantifier induced)
presupposition) sentence \textbf{Q'(P')} iff\textbf{ Q(X)} entails
\textbf{Q'(P')} for all properties \textbf{X}.



Notice that since in the above definition entailment should hold for
all properties, and thus also for the negation of the property
\textbf{P}, the
QI-presupposition is also a \emph{classical} presupposition.


Before various classes of quantifiers are analysed from the point of
view of presuppositions to which they give rise, notice the following
proposition:

\begin{pr}
If a sentence $S$ QI-presupposes a sentence $T$ and $T$ entails a sentence $U$,
then $S$ QI-presupposes $U$.
\end{pr}

Our purpose is now to consider various classes of quantifiers and
determiners in their relation to QI-presupposition. The following
proposition guarantees the existence of QI-presuppositions for increasing
monotonic (with respect to the second argument or \textbf{PMON})
determiners (which are
supposed to be conservative):
\begin{pr}
If \textbf{D $\in$ PMON} then \textbf{D(S)(P)}
QI-presupposes \textbf{D(S)(S)}
\end{pr}

\emph{Proof} Consider the sentence \textbf{D(S)(X)}. By conservativity it is
equivalent to \textbf{D(S)(S $\cap$ X)}, which by monotony entails
\textbf{D(S)(S)} for
every \textbf{X}. So \textbf{D(S)(P)} QI-presupposes \textbf{D(S)(S)}.

Since sentence \textbf{D(S)(S)} is not trivially true for determiners
\textbf{D} which
are weak, proposition 1 defines a class of non-trivial QI-presuppositions
for monotonic imcreasing determiners. Among them are well known
determiners giving rise to existential presuppositions such as those from
which definite descriptions and possesive clauses are formed.
It is possible to distinguish two other classes of determiners which,
although non monotonic, give also rise to QI-assertions. These are classes
\textbf{GEXT} of generalized existential determiners and \textbf{EXPT}
of (positive) exception
determiners (which is a subclass of generalized universal quantifiers,
cf. Keenan 1993).
 Concerning \textbf{GEXT} we will use the following property:


Prop 1: If\textbf{ D $\in$ GEXT} then for all \textbf{S, P, D(S)(P)} is
true only if
\textbf{S$\cap$P$\neq$$\emptyset$}

For the class \textbf{GEXT} we have the following presuppositional property:
\begin{pr}
If \textbf{D $\in$ GEXT} then \textbf{D(S)(P)}
QI-presupposes \textbf{ SOME(S)(S)}
\end{pr}

Notice furthermore, given Prop1, that generalized existential quantifiers
with exception clauses like \textbf{No...except  Bill} or \textbf{No...
except n} have more
specific QI-presuppositions which entail \textbf{SOME(S)(S)}. They are
specified
in the following propositions:
\begin{pr}
Sentence \textbf{ No(S) except n are (P)} QI-presupposes \textbf{ n(S)(S)}
\end{pr}
\begin{pr}
Sentence \textbf{No(S) except  Bill (is) (P)}
QI-presupposes \textbf{Bill is S}
\end{pr}

Let us consider now QI-presuppositions related to positive exception
determiners or rather to generalized universal quantifiers
interpreting nouns phrases with exception clauses. Although there are many
families of such noun phrases, differing by the exception complement in
the exception clause, I will consider
 here only the case of exception clauses
with the proper name \emph{PrN} as the exception complement; other cases
can be handled similarly. First, the following property indicates the
semantics of sentences with such noun phrases:

Prop 2: The sentence \textbf{All (S) except PrN (is)(P)} is true
iff \textbf{S-P=setPrN},
where \textbf{setPrN} is the element denoted by the proper name\textbf{ PrN}.
>From this property follows the following proposition:
\begin{pr}
The sentence \textbf{All(S) except PrN (is)(P)} QI-presupposes \textbf{PrN
(is)(S)}
\end{pr}
Thus \emph{All logicians except Bill are happy} QI-presupposes
\emph{Bill is a logician}

\subsection{Quantifier induced assertion}
Let us now define QI-assertion (quantifier induced assertion):

 (D2) Sentence \textbf{D(S)(P)} QI-asserts sentence \textbf{D'(S')(P)}
 iff\textbf{ D(S)(X)}
 entails \textbf{D'(S')(X)} for all \textbf{X}

 Again this definition as in the case of QI-presupposition is a
 generalization of the classical notion of assertion since from the fact
 that entailment should hold for all properties \textbf{X} it follows
 that it holds
 also for the negation of the property \textbf{P}.
 Clearly any sentence QI-asserts itself. Furthermore, we have also the
 following properties of QI-assertions:
 \begin{pr}
 If a sentence $S$ QI-asserts a sentence $T$ and $T$ QI-asserts a
 sentence $U$ then $S$ QI-asserts $U$.
 \end{pr}
 \begin{pr}
 If a sentence $S$ QI-asserts a sentence $T$ and $T$ QI-presupposes $U$
 then $S$ QI-presupposes $U$.
 \end{pr}

 Thus  QI-assertion is transitive and any QI-presupposition of a
 QI-assertion is a QI-presupposition of the asserting sentence.

 The case of QI-assertion needs a similar development as
 QI-presuppositions since the definition (D2) does not guarantee the
 existence of QI-assertions. It can be shown, however, that in this case
 it is also possible to distinguish various properties of determiners
 which give rise
 to particular QI-assertions.

 As a  first case we will consider persistent (i.e.
 momotomic with respect to the frist argument) determiners for which one
 has the following proposition:
 \begin{pr}
 If \textbf{D $\in$ PERS} then \textbf{D(S)(P)} QI-asserts
 \textbf{D(S')(P)} for all\textbf{ S'} such
 that \textbf{S $\subseteq$ S'}
 \end{pr}
 Thus \emph{A young denstist is sleeping} QI-asserts \emph{A dentist
 is sleeping}.

 Proposition 9 concerns a large sub-class of generalized existential
 quantifiers, roughly speaking those which do not contain ``exception
 clauses''.One  notices, however, that the whole class of \textbf{GEXT}
 quantifiers, even those which are not persistent (which contain an
 exception clause) give rise not only to QI-presuppositions but also to
 QI-assertions. Limitimg ourselves to those which have the exception
 complement in the form of a proper name we get the following proposition:
 \begin{pr}
 Sentence \textbf{No(S) except PrN (is)(P)} QI-asserts \textbf{PrN (is)(P)}
 \end{pr}
 The sub-class of universal generalized quantifiers which have been
 denoted by \textbf{EXPT}  above give also rise to particular
QI-assertions. They
 can easily be deduced from the definition of QI-assertion and the
 statements indicating the semantics of sentences with such
 constructions. The following proposition specifies such assertions for
 two families of universal generalized quantifiers:
 \begin{pr}
 A sentence of the form \textbf{All(S) except Bill (are)(P)} or of the form
 \textbf{All(S) except  n  (are)(P)} QI-asserts\textbf{ not-All(S)(are)(P)}
 \end{pr}
 To conclude this section I want to mention that there is another class
 of complex determiners which also give rise to specific QI-presuppositions
 and QI-assertions: these are determiners formed by \emph{inclusion
 clauses} of the type \emph{including NP} which interpret complex NP's
 like \emph{Most students, including Bill} or \emph{Some logicians
 including five from Albania}. Their asserted and presupposed information
 can be analysed in a quite similar way. For instance \emph{Most students,
 including Bill, drink} QI-presupposes \emph{Bill is a student} and
 QI-asserts \emph{Most students drink}.

 \section{Generalized categorial information}
 Our goal is now to generalize the notions of presupposition and
 assertion in such a way that we could speak about presupposition and
 assertion of (theoretically) \emph{any} category. More precisely, we
 want to say that an expression of a category C can have a (generalized)
 presupposition and a (generalized) assertion of the same category C.
 To do this we need the following preliminary observations.

 As we have seen in the preceeding sections, QI-assertions and
 QI-presuppositions are just particular entailments; they are entailments
 which should hold between a family of sentences (having, roughly speaking,
 the same subject and in which the VP varies) and a particular
 sentence. So first we have to generalize the notion of entailment in
 such a way that we could say that it holds between two expressions of a
 given category C. This generalisation has already been done: the needed
 relation is just the Boolean order in the Boolean algebras formed by the
 denotations of the category C (cf. Keenan and Faltz 1985).
 Since a priori not all expressions of a given category presuppose and
 assert we need to consider only Boolean algebras of a particular type:
 those which are used to interpret presupposing and asserting
 expressions. I will argue that expressions formed by a
 \emph{modifier} presuppose and assert and so we will be mainly concerned by
 restricting algebras (which are a special case of factor algebras) since
 precisely these algebras are used to interpret expressions with modifiers.
 Let me recall or introduce briefly the above notions.

  Modifiers are
 expressions which combine with ones in category C to form ones in
 category C. So they have the category $C/C$ for various choices of C.
 Thus semantically modifiers are interpreted by functions from an algebra
 (interpreting the category C) into itself. Algebras of such functions
 will be denoted by \textbf{CfC}.Again, as in the case of determiners,
 not full
 algerbas \textbf{CfC} are necessary for the semantic interpretation of natural
 language expressions. For instance, concerning (extensional)
 adjectival and adverbial modifiers one can suppose (cf. Keenan and Faltz
 1985)  that the set \textbf{F} all functions which
 interpret such modifiers are (positively) restricting. We will suppose
 that functions interpreting all types of modifiers (not only adjectival
 or adverbial) are restricting in the following
 sense:\textbf{ F(a) $\leq$ a}, where \textbf{F $\in$ CfC} and \textbf{a
$\in$ C}.
 All restricting functions \textbf{rf} from a Boolean algebra\textbf{B}
 to itself form a
 Boolean algebra called a
 (positive) restricting algebra which will be denoted by \textbf{BrfB}.

 Restricting algebras are a special case of factor algebras which are
 defined as follows. Let \textbf{B} be a Boolean algebra and \textbf{a} a
non-zero
 element of \textbf{B}. Then the set \textbf{B/a} defined
 as             $$B/a=\{x: x\leq a\}$$ is a
 Boolean algebra, where the zero element, meet and join are
 as in\textbf{ B}, the unit is \textbf{a} itself and the
 complement \textbf{cpl x=a $\cap$ non x}
 (where \textbf{non x} is the complement in \textbf{B}).
 \textbf{B/a} is called the factor algebra (of \textbf{B}) relative to
 (or generated by) \textbf{a}.
 The element \textbf{a}to which an algebra is relativised determines a relative
 negation and as such will determine a non-trivial presupposition.

 Consider now a Boolean algebra \textbf{BfB}, for an arbitrary
 \textbf{B}. Then the
 factor algebra \textbf{(BfB)/id} is just the restricting algebra
\textbf{BrfB},
 where \textbf{id} is the identity function.
 An important sub-class of intersecting functions for
 our purpose is
 constituted by so-called \emph{intersecting functions}. They are
 defined as follows:

 (D 3) \textbf{f $\in$ BfB} is \emph{intersective} iff \textbf{f(a)=a $\cap$
 f(1)}, for all
 \textbf{a $\in$ B}.

 The set of all intersective functions from a Booelan algebra into itself
 forms a Boolean algebra.
 Intersective functions were studied by Keenan (1983). He also showed
 that they interpret some adjectives (so-called \emph{absolute
 adjectives}) and relative clauses. It can also be argued that they
 can be used for the semantic description of \emph{positive implicative
 verbs} (for instance \emph{manage to escape} imples (in fact
 asserts) \emph{to escape} and of \emph{NP's with inclusion clauses}. I
will come back to
 this problem later on.

 Restrictive and intersective functions have their \emph{negative}
 counterparts: these are \emph{anti-restrictive} and their important sub-class
 \emph{anti-intersective} functions (cf. Zuber, forthcoming):

 (D4) A function \textbf{f $\in$ BfB} is \emph{ant-restrictive}
 iff \textbf{f(a) $\leq$
 n-a}, for all \textbf{a $\in$ B} (where \textbf{n-a} is the complement of
 \textbf{a} in \textbf{B}).

 (D5) A function \textbf{f $\in$ BfB} is \emph{anti-intersective} iff the
 following holds: $$\textbf{f(a)= n-a$\cap$ f(0)}$$

 Anti-restrictive functions are also related to factor algerbas: they
 correspond to factor algebras relative to anti-identity function
 \textbf{ant-id} (i.e. \textbf{f(x)= n-x}).

 Anti-intersective functions also form a Boolean algebra, a sub-algebra
 of the coresponding anti-restrictive algebra. They can be used
 to interprete semantically negative modifiers
 such as for instance negative implicative verbs (\emph{forget to close 
 the door} implies \emph{not to cloase the door}) some aspectual verbs
 and NP's with exception clauses.

 Now we are in a position to formulate the last observation concerning
 the form of linguistic expressions which can be said to assert and to
 presuppose: we can say, even if in some cases this is not obvious (cf.
 Proposition 2), that all presupposing and asserting expressions  are
 complex expressions of the form \textbf{M(E)} - which will be called
 \emph{modified expressions}, where \textbf{M} is a
 modifier which is interpreted
 by an intersective or an anti-intersective function. The case of
 relative clauses (and thus, in logical terminology, of definite
 descriptions) was analysed from
 this point of view in Keenan and Faltz (1985. p.262). One can claim also (cf.
 Moltmann 1995) that exception
 NPs and inclusion NPs considered above are formed by a modifier (like
 for instance \emph{except Bill} or \emph{including Mary} applied to some
 other noun noun phrases (or rather determiners).

 Given the above observations and the definition D1 of QI-presupposition
 we can now define a (generalized) presupposition:

  (D6) Expression E (of the category C) \emph{g-presupposes} (or for
  short \emph{presupposess}) the expression F
  (of the same category C) iff E is a modified expression of the form M(A)
  which denotes \textbf{a$\cap$f(1)}, where \textbf{a} is the denotation
  of A and \textbf{f} is an intersective ( or anti-intersective) function
  interpreting the modifier M, and F denotes \textbf{f(1)} (or \textbf{f(0)})

  One can show that the above definition is a generalization of the
  definition D1 which can be said to define, strictly speaking,
  a g-presupposition of a NP
 since in the presupposing sentence the VP does not contribute
  to the semantic content and the presupposed sentence can be transformed
  into an NP with a similar relevant content. For instance \emph{Bill is a
  student} can be transformed into \emph{Bill who is a student}.What is
  important is the fact that  the presupposing expresion contains an
  expression which can be considered as a variable and of which the
  content of the presupposed expression is independent.

  The definition of generalized assertion can be given in a similar way.
  Roughly speaking the assertion of a modified expression will be
  either the expression to which
  the positive modifier aplies (in the case when one uses an
  intersective function
  to interpret the modifier) or the negation of this
  expression (in the case when the modifier is interpreted by
  an anti-intersective function).

  We can now say that for instance the determiner \emph{All...except Bill}
  asserts \emph{not all} and presupposes \emph{Bill, who is a...}.
  Smilarly \emph{Most..., including Sue} asserts\emph{ Most} and
  presupposes \emph{Sue, who is a...}. Similarly the common noun
  \emph{poetess}presupposes \emph{woman} and asserts \emph{poet}.
  Many other, more familiar cases of assertion and presupposition can be
  analysed in the same way.. Furthermore, some general properties,
  similar to those indicated in the section 2 but concerning the generalized
  notion of assertion and presupposition can also be proved.
  \section{Conclusion}
  The existence of various constraints on functions used to interpret
  semantically linguistic expressions gives rise to a specific
  information determined by particular constraints. I have given some
  examples of the constraint giving rise to such function induced
  information and, interestingly enough, it appers that functions having the
  intersectivity property determine information known as presupposition
  and assertion. However the status of these two notions is here more general
  since they can be defined for expressions of various categories in
  which a (positive or negative)
  modifier occurs.

  By the same token, it has been shown that presuppositions and assertions
  are just generalized entailments satisfying some additional conditions.
  Consequently there is no need for non-standard logical systems such as
   default logic, multivalued logic or logic with truth-value
  gaps in order to account for various phenomena giving rise to assertions
  and presuppositions. As has been shown, a rather simple and natural
  framework  of
  Boolean algebras is sufficient to account for a huge class of data
  implied in the study of these notions.

  There are some arguments showing that the distinction between
  function-induced information and other types of information may be
  related to the distinction between grammatical and lexical information.
  Its study was limited here mainly to the study of modifier interpreting
  functions and to the related intersective and anti-intersective
  Boolean algebras. Clearly information induced by other types of
  functions related to specific linguistic phenomena is in need of a
  similar study. It is not obvious, however, that this can be done just
  by simple extension of the ideas presented here.


 \begin{thebibliography}{99}


  	\bibitem{}{Keenan, E.L. (1983) Boolean Algebra for
  	linguists, \emph{Working Papers in Linguistics}, UCLA}

  	\bibitem{}{Keenan, E.L. (1993) Natural Language, Sortal Reducibility and
  	Generalized Quantifiers, \emph{J. of Symbolic logic} 58-1, p.314-325}
  	

  	\bibitem{}{Keenan, E.L. and Faltz, L.M. (1985) \emph{Boolean Semantics
  	for Natural
  	Language,} D. Reidel Publishing Company, Dordrecht}
  	

  	\bibitem{}Moltmann, F. (1995) Exception sentences and polyadic
  	quantification, \emph{Linguistics and Philosophy}
  	

  	\bibitem{}Zuber, R. (forthcoming) On anti-restrictive algebras
  	
  \end{thebibliography}

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